# Numerical Simulation of Ammonothermal Crystal Growth of GaN—Current State, Challenges, and Prospects

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Functionality of the Ammonothermal GaN Growth Process

_{4}Cl [40] or NH

_{4}F [4]. Besides high-temperature versions of ammonoacidic growth, ammonobasic growth also operates in the retrograde solubility range [9], albeit at lower absolute temperatures and higher pressures than ammonoacidic growth.

## 3. Simulations of Fluid Flow and Temperature Field

#### 3.1. Simulation Domain and Geometry

#### 3.2. Axisymmetric 2D Calculations versus 3D Calculations

#### 3.3. Boundary Conditions

#### 3.4. Physics and Models Thereof

#### 3.5. Discretization in Space and Time

#### 3.6. Results

## 4. Simulations of the GaN Crystal Growth Process

## 5. Approaches to Validation

## 6. Open Questions That May Affect the Accuracy of Simulation Results

#### 6.1. Fluid Properties

_{c}< T < 1.2 T

_{c}, and pressures are usually in the range of 1.01 p

_{c}< p < 1.5 p

_{c}[100,101], with T

_{c}and p

_{c}representing the critical temperature and critical pressure of the fluid, respectively. For ammonia, this range would be 133.7 °C to 158.9 °C and 11.4 MPa to 17.0 MPa, which is far from the parameter range of ammonothermal growth of GaN [46]. This is illustrated in Figure 10b, which shows the pressure as a function of specific volume for different temperatures.

_{3}, N

_{2,}and H

_{2}[102]. Pimputkar et al. have studied this decomposition reaction in a combined numerical and experimental approach to determine an accurate description for the equilibrium constant for the ammonia decomposition reaction as a function of pressure and temperature and verified it against experimental data [102]. The determined equilibrium constant as a function of inverse temperature is shown in Figure 11a. For selected fill densities, the calculated mole fractions of ammonia in equilibrium as a function of temperature are depicted in Figure 11b alongside experimental data. Depending on the materials of the pressure-bearing materials, hydrogen may leave the otherwise closed system by diffusion [102]. When considering the composition of the ammonothermal reaction medium during actual GaN growth experiments, it is important to be aware that different ammonothermal growth environments may cause vastly different kinetics of ammonia decomposition. In other words, any ammonia mole fraction from 1.0 down to the equilibrium value may be present at some point in time during a growth experiment, and there is also a possibility that equilibrium is never reached. The kinetics of ammonia decomposition can be expected to depend heavily on the presence or absence of materials that can act as a catalyst for the decomposition reaction. Specifically, Ni is known to catalyze ammonia decomposition [107]. GaN growth is typically conducted in autoclaves made from nickel-base [9,20,44,48] and sometimes molybdenum-base alloys [47]; however, the autoclave wall is not necessarily in direct contact with ammonia. In order to prevent corrosion of the autoclave wall, as well as in order to minimize the introduction of transition metal impurities, hermetically sealed [108] or pressure-balanced [3,109] liners or capsules of different, more corrosion-resistant materials are often applied. Depending on the mineralizer, platinum [9,40,106], silver [19], or molybdenum [70] are used as liner materials for bulk GaN growth. Besides the catalytic properties of the inner wall, there is also a possibility that the mineralizer itself may affect ammonia decomposition [46]. Given that chlorine is known to poison catalysts of ammonia synthesis and decomposition [110,111,112], NH

_{4}Cl (and possibly further acidic mineralizers) might hinder ammonia decomposition [46]. In conclusion, it is not fully clarified how quickly ammonia decomposes under specific growth conditions of GaN. Consequently, it is not always clear which mole fractions of ammonia, nitrogen, and hydrogen should be assumed. To the author′s knowledge, there are also no numerical studies that account for the presence of nitrogen and hydrogen. Moreover, the sensitivity of simulation results to changes in ammonia, nitrogen, and hydrogen mole fractions has not been investigated yet.

_{4}Cl mineralizer, optical in situ measurements have shown that the optical transparency of the fluid decreases rapidly as temperature is increased, which is ascribed to increasing concentrations of solutes [87]. Though no measurements of transparency of infrared radiation have been reported, significant absorption of the fluid cannot be excluded and may be specific to the chemical species present.

_{4}Cl mineralizer [87]. However, Alt et al. do not comment on the reproducibility of their experiments, and the temperature deviations cannot unambiguously be assigned to a possible change of heat capacity.

#### 6.2. Possible Relevance of Solutal Buoyancy

_{3}). For lack of data on Ga concentration differences under actual growth conditions, we utilize data on the solubility limit of Ga, which were obtained under ammonoacidic conditions using NH

_{4}F mineralizer [39]. While the effect of solutal buoyancy may be smaller in reality (as there is likely no region in the autoclave with zero concentration of dissolved Ga), this consideration should be sufficient to elucidate whether solutal convection is likely to be of relevance. The results of this estimation are presented in Table 2. Accordingly, solutal buoyancy may very well have a significant impact, as the density increase due to Ga-containing solutes reaches the same order of magnitude as the density difference of pure ammonia that is induced by thermal gradients.

#### 6.3. Solubility of the Metal

#### 6.4. Dissolution and Growth Kinetics

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Basic functionality of ammonothermal GaN bulk growth process. Recrystallization of GaN takes place in an autoclave using a temperature gradient to create a driving force for dissolution and crystallization in different zones of the autoclave. Mass transport of Ga is achieved by the formation of soluble intermediates and their transport by natural convection of the fluid.

**Figure 2.**Schematic representation of the temperature program used for ammonothermal growth of GaN in the retrograde solubility range. The externally controlled temperature program (set temperatures) is based on [41]. The mean internal temperatures are not known but intended to indicate that there is a difference between internal and external temperatures, with the internal gradient likely being smaller than the externally applied gradient due to convective heat exchange between the two zones. The round inset visualizes that local fluid temperatures are often unstable and may not even follow a strictly oscillatory pattern, on the basis of [43] and references therein.

**Figure 3.**Setup for bulk crystal growth of GaN using an ammonothermal method. (

**a**) Complete setup consisting of autoclave with head assembly in a furnace, which are usually surrounded by some enclosure filled either with ambient air or nitrogen. The illustration assumes retrograde temperature dependence of solubility (in case of normal solubility, the locations of GaN seeds and nutrient would be inverted). Thick blue lines represent possible choices for the boundaries of the simulation domain. Parts drawn with gray line color (burst disc, insulation of autoclave head, liner) are optional and may not be used, depending on the mineralizer and safety concept. (

**b**–

**e**) Baffle geometries considered in simulation literature: (

**b**) ring-shaped baffle directly attached to the autoclave wall, (

**c**) ring-shaped baffle with a gap between baffle and autoclave wall, (

**d**) funnel-shaped baffle with a gap between baffle and autoclave wall.

**Figure 4.**Boundary conditions for different choices of the simulation domain. (

**a**,

**b**): fixed heater power densities; (

**a**) allows for gas exchange via open boundaries as well as radiative losses to the ambient, whereas (

**b**) uses a fixed temperature of the domain boundary, (

**c**,

**f**) heater-long constant fixed temperatures and adiabatic walls elsewhere, (

**d**,

**g**), fixed temperature distributions at all walls, and (

**c**,

**d**) show simulation domain including the autoclave wall, whereas (

**f**,

**g**) represent simulation domain excluding the autoclave walls.

**Figure 5.**Setup (

**a**) and simulation results (

**b**,

**c**) by Chen et al. for a reactor with an inner diameter of 2.22 cm and baffle opening of 10%: dimensionless temperature field (

**b**) and average velocity in the central hole and ring gap (

**c**). Reprinted with permission from Springer Nature Customer Service Centre GmbH: Springer Nature Research on Chemical Intermediates [52], Copyright 2011.

**Figure 6.**Effects of crystal size increase in a 2D simulation by Masuda et al. (

**a**) Considered experimental setup and chosen axisymmetric model thereof, (

**b**) temperature distribution in the vicinity of the seed and above the baffle, (

**c**) schematic of the two types of convection patterns that were found to develop for crystal radii below and above about 15 mm, respectively. Reprinted with permission from [25]. Copyright (2016) The Japan Society of Applied Physics.

**Figure 7.**Comparison of the effects of different thermal boundary conditions applied to the outer autoclave walls. (

**a**,

**d**) show temperature and flow fields of a global field simulation including the furnace and its surroundings, (

**b**,

**e**) temperature and flow fields with heater-long fixed wall temperatures and otherwise adiabatic walls. (

**c**,

**f**) have been obtained using wall temperature distributions extracted from a global field simulation including the furnace and its surroundings with a simulation domain limited to the autoclave. In all cases, three seed crystals and a ring-shaped baffle were considered in an axisymmetric 2D simulation. Major arrows are normalized and serve as a guide to the eye for better visibility. Reprinted from [57].

**Figure 8.**Properties of supercritical ammonia at 426.08 °C as a function of fill level (data from National Institute of Standards and Technology (NIST) database [98]).

**Figure 9.**System pressure for ammonia in a closed system as a function of temperature for fill levels from 10 to 100%, with fill levels referring to three different reference temperatures (reprinted from [46]). The reference temperatures correspond to the boiling point, a typical temperature for the introduction of liquified ammonia using a pressurized system, and room temperature, respectively [46].

**Figure 10.**(

**a**) Phase diagram of pure ammonia with contour lines of density (mol/l) and superimposed phases present at equilibrium. Calculated and extrapolated beyond 700 K using the reference multiparameter equation of state (MEOS) as provided by the National Institute of Standards and Technology (NIST). Reprinted from [102], Copyright 2016, with permission from Elsevier. (

**b**) Pressure as a function of specific volume of ammonia for selected temperatures (reprinted from [46], data from NIST database [98]). The hatched area in red marks the typical application range of supercritical fluids whereas the dark blue region approximately indicates the parameter range of ammonothermal GaN growth.

**Figure 11.**Data on ammonia decomposition at conditions relevant to ammonothermal GaN growth calculated by Pimputkar et al.: (

**a**) equilibrium constant K

_{p}for the ammonia decomposition reaction as a function of inverse temperature calculated for various total system pressures ranging from 1 to 300 MPa., (

**b**) calculated equilibrium mole fraction of ammonia as a function of temperature and initial fill density, overlaid with three experimentally determined data points. Both reprinted from [46], Copyright 2016, with permission from Elsevier.

**Table 1.**Process parameters of hydrothermal and ammonothermal growth. In the case of the ammonothermal method, the reports with the highest growth rates as of 2018 were chosen. The temperature dependency of solubility refers to the temperature range used in the growth process. T

_{CZ}refers to the crystallization zone temperature, ΔT to the temperature difference between growth and dissolution zones. Reprinted from [46].

Hydrothermal | Ammonothermal | |||||
---|---|---|---|---|---|---|

Material | Quartz | ZnO | GaN | |||

Process Route | Mineralizer-Free [103] | Low-Pressure Process [104] | High-Pressure Process [104] | [105] | Acidic [106] | Basic [3] |

T_{CZ}/°C | 445–500 | 345 | 360 | 300–430 | 625 | 575 |

T_{CZ/}T_{c} | 1.19–1.34 | 0.92 | 0.96 | 0.80–1.15 | 4.73 | 4.35 |

ΔT/°C | 25 | 10 | 25 | 10–20 | 50 | 30–45 |

Solubility | retrograde | normal | normal | retrograde | ||

p/MPa | 60–110 ^{1} | 70–100 | 100–150 | 70–255 | 80–150 | 250 |

p/p_{c} | 2.71–4.98 | 3.17–4.52 | 4.52–6.79 | 3.17–11.54 | 7.08–13.27 | 22.12 |

Mineralizer | none | Na_{2}CO_{3} | NaOH | NaOH | NH_{4}F | Na |

[0001] growth rate/µm/day | 0.3–2 | 400 | 1000 | 300 | 410 | 344 |

^{1}Based on the reported fill level.

**Table 2.**Estimate of temperature-induced density differences (labeled “supercritical NH

_{3}”) between growth and dissolution zone, and solute-induced density differences (labeled “dissolved Ga”).

Supercritical NH_{3} | Dissolved Ga | |
---|---|---|

Density difference/mol/L | 1.2–5.8 [44] | 0.5 [39] |

Density difference/g/L | 20.4–104.6 | 34.9 |

Mineralizer | Experimental Conditions | Range of Solubility Data | Reference |
---|---|---|---|

NH_{4}Cl/NH_{4}I mixture | 450–550 °C (external) 96–102 MPa 0.42–0.51 mmol NH _{4}X/mL (*)100 h | 0.048–0.052 mol GaN/mol NH_{4}X (*)0.15–1.2 mol% 0.42–0.47 mmol/mL (*) | D. Tomida [117] |

NH_{4}Cl/NH_{4}Br mixture | 450–550 °C (external) 96–103 MPa 0.40–0.51 mmol NH _{4}X/mL (*)100 h | 0.11–0.12 mol GaN/mol NH_{4}X (*)0.35–1.23 mol% 0.40–0.51 mmol/mL (*) | D. Tomida 2018 [117] |

NH_{4}Cl | 200–550 °C (external) 67.7–100.9 MPa 0.33–3.30 mmol NH _{4}Cl/mL (*)120 h | 0–2.4 mol GaN/mol NH_{4}Cl0–7.04 mol% (*) 0–7.92 mmol/mL (*) | D. Ehrentraut 2008 [122] |

420–600 °C (external) 55–150 MPa 0–4.04 mmol NH _{4}Cl/mL (*)100 h | up to 0.41 mol GaN/mol NH_{4}Cl (*)0.04–5.47 mol% 0.01–1.65 mmol/mL (*) | D. Tomida 2010 [118] | |

NH_{4}F | 486–572 °C (internal) 16–175 MPa 0.76 mmol NH _{4}F/mL Until observation of saturation | 0–0.11 mol GaN/mol NH_{4}F0–1.03 mol% 0–0.08 mmol/mL | S. Schimmel 2017/2018 [56]/[46] |

Na | 415–650 °C (internal) 200 MPa 14.13–21.89 mmol Na/mL (*) 45–316 h | 0.00017–0.00122 mol GaN/mol Na (*) 0.02–0.12 mol% 0.07–3.45 mmol/mL (*) | S. Griffiths 2016 [94] |

NaNH_{2} | 450–650 °C (external) 76 ± 12 MPa 0.14 mmol NaNH _{2}/mL (*)120 h | up to 0.16 mol GaN/mol NaNH_{2} (*)up to 2.44 mol% (*) up to 0.02 mmol/mL (*) | T. Hashimoto 2007/2011 [123,124] |

NaN_{3} | 396–538 °C (internal) 259–268 MPa 0.65 ± 0.07 mmol NaN _{3}/mL Until observation of saturation | 0.02–0.05 mol GaN/mol NaN_{3}0.04–0.15 mol% 0.01–0.04 mmol/mL | S. Schimmel 2017/2018 [56]/[46] |

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Schimmel, S.; Tomida, D.; Ishiguro, T.; Honda, Y.; Chichibu, S.; Amano, H.
Numerical Simulation of Ammonothermal Crystal Growth of GaN—Current State, Challenges, and Prospects. *Crystals* **2021**, *11*, 356.
https://doi.org/10.3390/cryst11040356

**AMA Style**

Schimmel S, Tomida D, Ishiguro T, Honda Y, Chichibu S, Amano H.
Numerical Simulation of Ammonothermal Crystal Growth of GaN—Current State, Challenges, and Prospects. *Crystals*. 2021; 11(4):356.
https://doi.org/10.3390/cryst11040356

**Chicago/Turabian Style**

Schimmel, Saskia, Daisuke Tomida, Tohru Ishiguro, Yoshio Honda, Shigefusa Chichibu, and Hiroshi Amano.
2021. "Numerical Simulation of Ammonothermal Crystal Growth of GaN—Current State, Challenges, and Prospects" *Crystals* 11, no. 4: 356.
https://doi.org/10.3390/cryst11040356