Beyond ab initio reaction simulator an application to GaN metalorganic vapor phase epitaxy A. Kusaba1S. Nitta2K. Shiraishi2T. Kuboyama3and Y. Kangawa1

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Beyond ab initio reaction simulator: an application to GaN metalorganic

vapor phase epitaxy

A. Kusaba,1S. Nitta,2K. Shiraishi,2T. Kuboyama,3and Y. Kangawa1

1)Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka 816-8580,

Japan

2)Institute of Materials and Systems for Sustainability, Nagoya University, Chikusa-ku, Nagoya 464-8601,

Japan

3)Computer Centre, Gakushuin University, Toshima-ku, Tokyo 171-8588, Japan

(*Electronic mail: kusaba@riam.kyushu-u.ac.jp)

(Dated: 24 October 2022)

To develop a quantitative reaction simulator, data assimilation was performed using high-resolution time-of-ﬂight mass

spectrometry (TOF-MS) data applied to GaN metalorganic vapor phase epitaxy system. Incorporating ab initio knowl-

edge into the optimization successfully reproduces not only the concentration of CH4(an impurity precursor) as an

objective variable but also known reaction pathways. The simulation results show signiﬁcant production of GaH3, a

precursor of GaN, which has been difﬁcult to detect in TOF-MS experiments. Our proposed approach is expected to be

applicable to other applied physics ﬁelds that require quantitative prediction that goes beyond ab initio reaction rates.

Crystal growth technology of III-nitride semiconductors

has been actively developed to fabricate optical and electronic

devices1–6. Reactor simulators that solve for heat and mass

transfer, chemical reactions, and phase transitions have played

a role in these developments7–12. The reaction model and its

kinetic parameters are key components of reactor simulators,

especially for the reaction system for chemical vapor deposi-

tion. It is generally quite difﬁcult to determine the complete

kinetics of complex gas-phase reaction networks using exper-

iments alone. Thus, such reaction systems have been studied

by ab initio calculations, e.g., density functional theory (DFT)

and transition state theory (TST)13–21. Increased semiconduc-

tor device performance has recently necessitated the use of

simulators for quantitative process optimization that goes be-

yond obtaining a qualitative understanding22–25. However, the

predictive performance of ab initio reaction rates is often in-

adequate for such quantitative requirements.

The aim of this study is to realize quantitative simulation

by data assimilation for a reaction model with rate constant

parameters for the GaN metalorganic vapor phase epitaxy

(MOVPE) system. A complex reaction network generally in-

volves many parameters, and experimental measurements are

costly and provide limited data. Under these circumstances,

two measures are taken to ensure that the solution is not indef-

inite or an overﬁt. First, a reduced reaction model is adopted

that excludes many radical reactions and consists of fewer re-

actions than conventional models21. This reduced model has

been demonstrated to reproduce the latest experimental fact

on the reaction pathway that the removal of the methyl group

in trimethylgallium (TMG) decomposition is caused by reac-

tions with NH326. Second, the optimization is performed with

the strategy that not only focuses on minimizing simulation

errors, but also maintains the theoretical basis from ab initio

calculations. In addition, the reaction pathway becomes a fun-

damental consideration in selecting a solution. This approach

is used to develop a reliable and quantitative reaction simula-

tor in this study.

Experimental data were obtained by high-resolution time-

of-ﬂight mass spectrometry (TOF-MS)26–28 as the detection

intensity of the individual molecules in MOVPE environ-

ments. This state-of-the-art mass spectrometric technique can

distinguish between NH2and CH4, which is not possible us-

ing conventional quadrupole mass spectrometry (QMS)29. In-

deed, this study relies on the CH4concentration data con-

verted from its intensity data. Details of the data conversion

are presented in Appendix A. Experimental measurements

were performed dozens of times at different temperatures, but

only data with kinetic information were used for data assimi-

lation.

The input–output relationship of the simulator is concisely

expressed as

ˆc(x) = f(Tset ;k).(1)

Here, the output ˆc(x)is the concentration distribution, the in-

put Tset is the heater temperature (an experimental condition),

the simulation parameter kis the reaction rate constant, and

the simulator fsolves the ordinary differential equations of

the reaction kinetics for our reduced reaction model shown in

Table I. The reaction rate constant is determined using DFT:

B3LYP/6-311G(d,p) and TST21. The modiﬁed Arrhenius for-

mat given below is used to model the temperature dependence

of the ab initio rate constant and subsequently tuned to better

reproduce the experimental data:

k= (qAA)Texp−(qEEact )

kbT.(2)

Here, the parameters Eact and Aare the activation energy and

preexponential factor obtained from ab initio calculations, and

qEand qAare tuning coefﬁcients for these parameters. Note

that Eact is not equal to the simple DFT activation energy be-

cause of the effects of TST. More details on the simulator can

be found in Appendix B.

A multiobjective optimization was performed to determine

the k(i.e., qEand qA) that maintains the theoretical basis ob-

tained from ab initio calculations and reproduces the experi-

mental data well. The ab initio ratios of Eact were used as the

theoretical basis. The concentration of CH4(an impurity pre-

cursor) was used as a measure of the simulation performance

arXiv:2210.11748v1 [cond-mat.mtrl-sci] 21 Oct 2022

because CH4is a stable molecule and thus, the corresponding

detection data are reliable and can be converted to concentra-

tion data by a reasonable scheme. Our optimization problem

is expressed as follows:

minimize









∑

i ˆcCH4(xd;Tset i,k)−cCH4(Tset i)

cCH4(Tset i)!2

N∑

jqEj−qEj2

.(3)

The ﬁrst objective function is based on the relative error in

the CH4concentration between the experimental (cCH4) and

simulation ( ˆcCH4) results at the detection position xd, where i

is the index of the experimental conditions. In this case, only

Tset was varied. The second objective function is the variance

of qE, where jis the reaction index and Nis the total number

of considered reactions. The smaller the variance is (i.e., the

closer the coefﬁcients qEjare to each other), the closer the

ratios of the modiﬁed activation energies (qEjEact j/qEkEact k)

are to the ab initio ratios (Eact j/Eact k). Multiobjective opti-

mization determines the best trade-off solutions (i.e., Pareto

solutions) rather than just one optimal solution. Evolutionary

algorithms are widely used for ﬁnding Pareto solutions to mul-

tiobjective optimization problems. A well-known algorithm

called the fast elitist nondominated sorting genetic algorithm

(NSGA-II) was employed in this study30.

Although the parameters are softly constrained by the sec-

ond objective function, the reaction model consisting of 29

reactions still has 58 parameters. To avoid difﬁculties in inter-

preting the tuning results, hard constraints are also imposed by

requiring qEto be the same for each reaction group (Groups

G1–G5, R7, and R14 in Table I). That is, the chemical equa-

tions belonging to the same group have the same reaction part-

ners (H2or NH3) and byproducts (CH4, NH3, or H2) and dif-

fer only in the substituents (−CH3,−NH2, or −H) not in-

volved in the reaction. Thus, these equations are expected to

have the same degree of error in the ab initio Eact . In addition,

as the preexponential factor does not have as large an inﬂu-

ence as the activation energy, the same qAwas assumed for all

reactions for simplicity.

Figure 1(a) shows the Pareto solutions obtained from

30,000 function evaluations in the optimization. Here, a

marker represents a set of tuning coefﬁcients, i.e., qEfor each

reaction group and qA. These approximate Pareto solutions

have the smallest variance of qEamong the many possible

sets of coefﬁcients resulting in the same error. The solutions

correspond to a trade-off between the two objective functions.

That is, the larger the variance of qEis, the lower the root

mean squared percentage error (RMSPE) of CH4concentra-

tion is. However, even with a solution that emphasizes low-

ering the variance of qE, the RMSPE is only less than 25%.

In Fig. 1(b), the horizontal axis is qEand the vertical axis is

the RMSPE. A marker in Fig. 1(a) corresponds to 7 markers

aligned horizontally in Fig. 1(b) at the same RMSPE level. A

solution with an RMSPE of approximately 25% has an almost

identical qEof approximately 0.85 for each reaction group,

that is, this solution maintains the ab initio ratios of Eact . The

solutions that emphasize reducing the RMSPE have different

FIG. 1. (a) Pareto solutions for two objective functions obtained by

NSGA-II. The ﬁrst objective function is expressed in terms of the

root mean squared percentage error (RMSPE). (b) The RMSPE de-

pendence of the qEcomponent for the different reaction groups of

the solutions shown in (a).

qEvalues for each reaction group. For example, the solution

with an RMSPE of 15.0% presented in Fig. 1(a) has a set of

qEs at a 15.0% RMSPE level in Fig. 1(b) (i.e., 0.795, 0.885,

0.846, 0.849, 0.848, 0.850 and 0.846 for G1, G2, G3, G4,

G5, R7 and R14, respectively). To reduce the RMSPE, the qE

for Group G1 needs to be relatively decreased, and the qEfor

Group G2 needs to be relatively increased. Note that qEEact

for all reactions is an absolute value that has been decreased

from the ab initio Eact to obtain these Pareto solutions. In ad-

dition, qAexhibited ﬂuctuations within 20% corresponding to

discontinuous changes in qE.

Figure 2 is a comparison of the simulation performance be-

tween the parameter sets tuned by the optimization and the

original, untuned parameter set. The difference in the ob-

served CH4concentration corresponds to the difference in the

experimental conditions Tset . Simulations were performed at

each Tset using a number of tuned parameter sets and the orig-

inal parameter set. The results obtained using the tuned pa-

rameter sets are plotted in different colors depending on the

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Beyondabinitioreactionsimulator:anapplicationtoGaNmetalorganicvaporphaseepitaxyA.Kusaba,1S.Nitta,2K.Shiraishi,2T.Kuboyama,3andY.Kangawa11)ResearchInstituteforAppliedMechanics,KyushuUniversity,Kasuga,Fukuoka816-8580,Japan2)InstituteofMaterialsandSystemsforSustainability,NagoyaUniversity,Chikusa-ku,Na...

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Beyond ab initio reaction simulator an application to GaN metalorganic vapor phase epitaxy A. Kusaba1S. Nitta2K. Shiraishi2T. Kuboyama3and Y. Kangawa1

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