# Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals

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## Abstract

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## 1. Introduction

## 2. Numerical Model

#### 2.1. Overview of Modelling Scheme

`FZone`[16], which is suitable only for the cylindrical phase, or a transient program

`FZoneT`[17], which describes the entire process: start cone, cylinder and end cone.

`FZone`and

`FZoneT`programs are used to calculate melt shape (box 3).

`FZoneT`also allows obtaining the melt volume dynamics (box 8): the history of the change in crystal diameter, melt volume, etc. More information about these programs is given in Section 2.2.

`OpenFOAM`[18] hydrodynamic simulation (box 6). This includes transient simulation of the melt flow, heat transfer and species transport; see Section 2.3. To obtain boundary conditions for the melt velocity and temperature at the free surface, a 3D inductor shape is used to calculate the induced heat and Lorentz forces [19] (boxes 4 and 5).

`OpenFOAM`hydrodynamic simulation produces the radial distribution of the species concentration $C\left(r\right)$ in the grown crystal and the effective segregation coefficient ${k}_{\mathrm{eff}}$. This data, together with the melt volume dynamics, is then used in the transient

`0D-segregation`program (box 9), which predicts the temporal evolution of the average species concentration in the melt [20]. The complete description of the mathematical model for

`0D-segregation`is presented in Section 2.4. Finally, the results of

`0D-segregation`program are interpolated on the point grid that is created in crystal domain (box 10); this process is schematically shown in Figure 4.

#### 2.2. Phase Boundaries

`FZone`, which has been used for the calculation of phase boundaries, is provided in [16]. The program considers axially symmetrical approximations of phase boundaries. The melting interface, crystallization interface and open melting front are moved according to the heat flux balance; the influence of the melt flow is neglected. The program

`FZone`operates in quasi-stationary approximation: it describes the cylindrical phase of the process, i.e., it calculates the shape of the phase boundaries for a process stage when they do not change in time.

`FZoneT`[17]. The evolution of the crystal diameter and melt volume is saved in a file, which can be later read by the

`0D-segregation`program. The species concentration inflow from the melting feed rod (the integral from Equation (3)) is also saved in a file. In the present work, the transient

`FZoneT`program was used to simulate the entire crystal growth (the start cone, cylinder and end cone), and the results from three time instants from the start cone stage were selected for melt flow simulations.

#### 2.2.1. Electromagnetic Field

`FZone`is used, the EM field is calculated in 3D using boundary elements and assuming negligible skin depth [19]. The EM field is iteratively recalculated until the shape of the phase boundaries converges. However, in the transient

`FZoneT`version, only an axially symmetrical (also denoted by 2D) EM field is simulated due to limited computational resources. In the 2D EM simulations, inductor slits are modelled by setting an artificial magnetic field source surface density [16], which allows part of the magnetic field lines to penetrate the inner part of the inductor.

#### 2.3. Species Transport in Melt

`OpenFOAM`hydrodynamic solver described in [22]. This assumes fixed phase boundaries and uses the transient, incompressible, laminar Navier–Stokes equation for melt velocity, with the Boussinesq approximation for the description of thermal convection. Boundary conditions for velocity are as follows:

- Marangoni force and the EM force are applied on the free melt surface.
- Fixed velocity (crystal pulling speed and rotation)—on the crystallization interface.

- On the crystallization interface: $D\frac{\partial C}{\partial n}={v}_{C}(1-{k}_{0})Ccos\theta ,$ where n is the normal coordinate, ${k}_{0}$ is the segregation coefficient, and $\theta $ is the the angle between the horizontal plane and the interface normal vector.
- On the melt free surface: $\frac{\partial C}{\partial n}=0$ due to the assumption of a pure gas atmosphere and lack of evaporation [24].
- On the melting interface: $C=1$ arb.u., i.e., the species concentration is normalized to the initial concentration in the feed rod, which is assumed to be homogeneous.

#### 2.4. 2D Species Distribution in Crystal

`OpenFOAM`simulations (see Section 3.3.1) and can depend on the process parameters, e.g., the crystal diameter ${D}_{C}$.

`Python`language. This program is called

`0D-segregation`, because it describes transient species segregation without spatial dimensions, i.e., disregarding spatial concentration distribution in the melt. The program source code is published in [20]. Two functionalities of the program are available:

- Importing the data about process dynamics (time-dependent ${V}_{m}$, ${C}_{F}$, $\mathrm{\Delta}{V}_{\mathrm{out}}$ and $\mathrm{\Delta}{V}_{\mathrm{in}}$) from transient phase boundary simulations with
`FZoneT`. - Creating an approximate description of the cone phase based only on the simplified crystal shape described as ${D}_{C}\left(L\right)$, where L is the crystal length:
- Due to the assumption of constant pulling velocity, $L\propto t$.
- Cone surfaces are approximated as having constant slope, and thus ${D}_{C}\left(t\right)\propto t$.
- The free surface height above the external triple point is assumed to be constant even during the cone phases because it is impossible to predict its evolution for an arbitrary crystal shape (without experimental data); therefore, ${V}_{m}\left(t\right)\propto {D}_{C}{\left(t\right)}^{2}$.
- The crystallized volume is proportional to the crystal cross-section: $\mathrm{\Delta}{V}_{\mathrm{out}}={v}_{C}{S}_{C}\mathrm{\Delta}t$, where ${v}_{C}$ is the crystal pulling velocity, and ${S}_{C}=\frac{\pi}{4}{D}_{C}{\left(t\right)}^{2}$ is the crystal cross-section area. Therefore, $\mathrm{\Delta}{V}_{\mathrm{out}}\propto {D}_{C}{\left(t\right)}^{2}$.
- Due to silicon mass conservation, $\mathrm{\Delta}{V}_{\mathrm{in}}=\mathrm{\Delta}{V}_{m}+\mathrm{\Delta}{V}_{\mathrm{out}}$.

## 3. Results

#### 3.1. Description of the Experiment

#### 3.2. Phase Boundaries

#### 3.2.1. Quasi-Stationary Simulations

`FZone`program to obtain the shape of phase boundaries for the system described in Section 3.1, which are later used in hydrodynamics simulations in Section 3.3. 3D EM simulations are performed to precisely describe the EM field created by the high-frequency inductor. An example of the results of 3D EM simulations (induced heat on silicon free surface) is shown in Figure 5.

#### 3.2.2. Influence of Three-Dimensionality of the EM Field

#### 3.2.3. Transient Simulations

`FZone`was applied in the present work for the cylindrical growth stage, the transient model should be considered for the initial and final stages of the process when the crystal or feed rod is short and the diameter is changing. Therefore, a transient simulation of the temperature field and phase boundaries has been conducted using

`FZoneT`, starting from a small cone with a diameter of 20 mm and a length of 12 mm at time $t=0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{i}\mathrm{n}$. A rather good agreement with the experiment was obtained for the shape of the phase boundaries during the cone and cylindrical growth stages; see Figure 8.

`FZoneT`could be caused by differences in input data (inductor current and feed velocity) or by model limitations (e.g., a fluid film model describing the melting of a macroscopically smooth open melting front).

`FZoneT`is plotted in Figure 9. It was verified that ${D}_{F}$ during the cylindrical growth stage was 90 mm both in the experiment (from process photos) and

`FZoneT`simulation in accordance with the steady-state mass conservation ${D}_{F}={D}_{C}\sqrt{{v}_{C}/{v}_{F}}$.

#### 3.3. Species Transport in Melt

`FZone`or

`FZoneT`and presented in the previous section, 3D melt flow simulations were run with

`OpenFOAM`using the model described in Section 2.3. The most important physical and numerical parameters are given in Table 2. An example of the simulation results for the carbon transport during the cylindrical phase is shown in Figure 10. Despite the non-symmetrical high-frequency inductor, the carbon concentration field is mostly axially symmetrical.

`OpenFOAM`simulations can be performed only using a fixed melt shape (stationary phase boundaries). The

`FZone`calculation of the cylindrical growth stage with a diameter of 102 mm was used. In addition, the results from

`FZoneT`with several cone diameters were selected to simulate the melt flow in a cone phase: 80, 60 and 45 mm.

#### 3.3.1. Effective Segregation Coefficient

`OpenFOAM`melt flow simulations, the effective segregation coefficient can be calculated:

`OpenFOAM`calculation cells.

#### 3.3.2. Increased Crystal Rotation Rate

#### 3.4. Species Distribution in Crystal

`0D-segregation`, which calculates the axial distribution of impurities (Section 2.4), was verified using an analytical solution for a theoretical “cylindrical crystal”—a crystal with constant diameter, disregarding the cone phases. The results of the verification are presented in Appendix B.

`FZoneT`program) is not available, the

`0D-segregation`program has a setting that creates a simplified crystal shape ${D}_{C}\left(L\right)$ as described in the end of Section 2.4. Figure 18 (left) shows this simplified shape for a 102 mm crystal: the crystal diameter and melt volume are functions of the crystal length and are compared with the shape simulated by

`FZoneT`.

`FZoneT`, was combined with radial distributions at corresponding diameters (shown in Figure 15), and linear interpolation was used between them. As a result, the 2D $C(r,z)$ distribution in the entire crystal is shown in Figure 19.

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**The time-dependence of the shape of the phase boundaries and global temperature field in Si plotted each 15 min. The spacing between isotherms is 50 K.

## Appendix B

`0D-segregation`for axial distribution of impurities (Section 2.4). The system consists of a crystal with a constant 102 mm diameter, ignoring the cone stages at the beginning and end of the growth process. The axial distribution of impurities in such a system can be described by an analytical formula [32]:

**Figure A2.**Comparison of model results with analytical formula for species concentration in the cylindrical crystal (without start and end cones) with different segregation coefficient values ${k}_{\mathrm{eff}}$ (L represents the crystal length, and H represents the height of the molten zone).

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**Figure 1.**Axially symmetrical scheme of the FZ method, with designations of the most important parts and the used coordinate system.

**Figure 2.**

**Left**—a simplified scheme of species distribution in the crystal ($z<0$) and the molten zone ($z>0$) near the interface ($z=0$).

**Right**—the dependence of the effective segregation coefficient ${k}_{\mathrm{eff}}$ on diffusion layer thickness d (e.g., [4]) for different species [16,17,18,19].

**Figure 3.**The overall scheme of the used numerical models (

**orange boxes**) and the data that is being exchanged between them (

**grey boxes**).

**Figure 4.**The scheme of species concentration interpolation from

`OpenFOAM`melt flow simulations of the cone stage and cylinder stage (

**a**) and

`0D-segregation`simulations (

**b**) on the grown crystal mesh (

**c**).

**Figure 5.**3D high-frequency electromagnetically induced heat sources on silicon surfaces used for 3D melt flow simulations. The one-turn inductor is shown in grey.

**Figure 8.**Comparison of the shapes of phase boundaries in the transient

`FZoneT`simulation with experimental photos during the cone growth (0–30 min) and cylindrical growth (40 min) stages.

**Figure 9.**Comparison of transient

`FZoneT`simulation results to the experiment during the entire growth process.

**Figure 10.**The time-averaged meridional melt velocity in the $xz$ plane, temperature in the $yz$ plane and carbon concentration on the crystallization interface, simulated for the cylindrical phase (${D}_{C}=102$ mm). The direction of the main inductor current suppliers corresponds to the x axis.

**Figure 11.**The time-averaged melt velocity (

**left**) and carbon concentration (

**right**) in the vertical cross-section of the melt, simulated for the cylindrical phase (${D}_{C}=102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$) and cone phase (${D}_{C}<102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$).

**Figure 12.**Absolute values of the EM and Marangoni force density on the free melt surface during the cylindrical phase (${D}_{C}=102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$).

**Figure 13.**The distribution of carbon (

**left**) and boron (

**right**) concentration in the melt, near the crystallization interface (${D}_{C}=102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$). The normal coordinate is denoted by n, and different radial positions r are considered. Boundary layer thicknesses from the BPS model (1) are shown with vertical black lines.

**Figure 14.**The time-averaged carbon concentration on the crystallization interface, simulated for the cone phase (${D}_{C}<102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$) and cylindrical phase (${D}_{C}=102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$).

**Figure 15.**Normalized radial distributions of carbon (

**left**) and boron (

**right**) concentrations in the grown crystal (yellow line for the cylindrical phase, violet and orange lines for the cone phase), using crystal rotation rates 6 rpm (solid lines) and 12 rpm (dashed lines).

**Figure 16.**The effective segregation coefficient ${k}_{\mathrm{eff}}$ of carbon (

**left**) and boron (

**right**) obtained for different crystal diameters ${D}_{C}$ and crystal rotation rates of 6 rpm (black) and 12 rpm (orange).

**Figure 17.**The time-averaged melt velocity in the vertical cross-section of the melt, simulated for the cylindrical phase with ${D}_{C}=102\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{m}$ and crystal rotation rates of 6 rpm (

**left**) and 12 rpm (

**right**).

**Figure 18.**The crystal diameter and melt volume as a function of the crystal length L for the considered ${D}_{C}=102$ mm system with cone stages (

**left**) and comparison of the axial carbon concentration in the crystal using different crystal shapes with ${k}_{\mathrm{eff}}\left({D}_{C}\right)$ from the melt flow simulation results (

**right**). Vertical grey lines represent the boundaries between the start cone, cylinder and end cone stages.

**Figure 19.**An example of the carbon and boron distribution in the grown crystal (

**top**) and the axial distribution of radially averaged concentration of these species (

**bottom**).

Parameter | Value |
---|---|

Crystal diameter ${D}_{C}$ | 102 mm (cylinder phase) |

Feed rod diameter ${D}_{F}$ | 90 mm |

Crystal pulling rate ${v}_{C}$ | 3.5 mm/min |

Feed rod push rate ${v}_{F}$ | 4.5 mm/min (cylinder phase) |

Crystal rotation rate ${\omega}_{C}$ | 6 rpm |

Feed rod rotation rate ${\omega}_{F}$ | −0.8 rpm |

Zone height ${H}_{Z}$ | 27 mm (cylinder phase) |

Inductor frequency f | 3 MHz |

Parameter | Value |
---|---|

Silicon density $\rho $ | $2580\phantom{\rule{0.166667em}{0ex}}\mathrm{k}\mathrm{g}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}{\mathrm{m}}^{3}$ |

Silicon viscosity $\eta $ | $8.6\xb7{10}^{-4}\phantom{\rule{0.166667em}{0ex}}\mathrm{P}\mathrm{a}\xb7\mathrm{s}$ |

Silicon heat conductivity $\lambda $ | $67\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{m}\xb7\mathrm{K}$ |

Silicon specific heat capacity ${c}_{p}$ | $1000\phantom{\rule{0.166667em}{0ex}}\mathrm{J}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{k}\mathrm{g}\xb7\mathrm{K}$ |

Silicon thermal expansion coefficient $\beta $ | ${10}^{-4}\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{K}$ |

Marangoni coefficient M | $-1.3\xb7{10}^{-4}\phantom{\rule{0.166667em}{0ex}}\mathrm{N}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{m}\xb7\mathrm{K}$ [26] |

Carbon diffusion coefficient D | $7\xb7{10}^{-9}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{s}$ [28] |

Carbon segregation coefficient ${k}_{0}$ | 0.07 [3] |

Boron diffusion coefficient ${D}^{\prime}$ | $1.2\xb7{10}^{-8}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathrm{s}$ [29] |

Boron segregation coefficient ${k}_{0}^{\prime}$ | 0.8 [29] |

Total number of mesh elements | 614,000 |

Largest element size | 0.8–1.4 mm (inside the melt) |

Smallest element thickness | 0.02–0.03 mm (at the crystallization interface) |

Time step | 2 ms |

Total simulation time | 350–500 s |

Averaging interval for $C,T,\overrightarrow{v}$ | 100 s |

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## Share and Cite

**MDPI and ACS Style**

Surovovs, K.; Surovovs, M.; Sabanskis, A.; Virbulis, J.; Dadzis, K.; Menzel, R.; Abrosimov, N. Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals. *Crystals* **2022**, *12*, 1718.
https://doi.org/10.3390/cryst12121718

**AMA Style**

Surovovs K, Surovovs M, Sabanskis A, Virbulis J, Dadzis K, Menzel R, Abrosimov N. Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals. *Crystals*. 2022; 12(12):1718.
https://doi.org/10.3390/cryst12121718

**Chicago/Turabian Style**

Surovovs, Kirils, Maksims Surovovs, Andrejs Sabanskis, Jānis Virbulis, Kaspars Dadzis, Robert Menzel, and Nikolay Abrosimov. 2022. "Numerical Simulation of Species Segregation and 2D Distribution in the Floating Zone Silicon Crystals" *Crystals* 12, no. 12: 1718.
https://doi.org/10.3390/cryst12121718