# StyleKQC: A Style-Variant Paraphrase Corpus for
Korean Questions and Commands

Won Ik Cho1, Sangwhan Moon2,3, Jong In Kim4, Seok Min Kim1, Nam Soo Kim1
Department of Electrical and Computer Engineering and INMC, Seoul National
University1
Department of Computer Science, Tokyo Institute of Technology2, Odd Concepts
Inc.3
Interdisciplinary Program in Cognitive Science, Seoul National University4
wicho@hi.snu.ac.kr,sangwhan<EMAIL_ADDRESS>
smkim<EMAIL_ADDRESS>

###### Abstract

Paraphrasing is often performed with less concern for controlled style
conversion. Especially for questions and commands, style-variant paraphrasing
can be crucial in tone and manner, which also matters with industrial
applications such as dialog system. In this paper, we attack this issue with a
corpus construction scheme that simultaneously considers the core content and
style of directives, namely intent and formality, for the Korean language.
Utilizing manually generated natural language queries on six daily topics, we
expand the corpus to formal and informal sentences by human rewriting and
transferring. We verify the validity and industrial applicability of our
approach by checking the adequate classification and inference performance
that fit with the fine-tuning approaches, at the same time proposing a
supervised formality transfer task.

##  1 Introduction

Paraphrasing, the act of using different sentences with the same meaning
Bhagat and Hovy (2013), is strongly related to the text style conversion or
transfer Yamshchikov et al. (2020). While prior studies often modify sentiment
or offensiveness Logeswaran et al. (2018); dos Santos et al. (2018), in view
of paraphrase, it should be well checked whether the core content of the
sentence is maintained during the conversion process. If the sentence meaning
stays the same while changing politeness or formality Rao and Tetreault
(2018), we can call it paraphrasing or rewriting. Such styles can be
represented in diverse ways across genre, domain, and language Jhamtani et al.
(2017); Fu et al. (2018); Yang et al. (2019).

We deal with the scheme of constructing a corpus of style-variant paraphrases
for directive sentences such as questions and commands, targeting the Korean
language where politeness (suffix) and honorifics play a significant role in
conversation Strauss and Eun (2005). Here, we consider topic and speech act as
attributes constituting the directive sentence Cho et al. (2020) and construct
a formal style paraphrase set using the natural language queries displaying
each topic and speech act. Finally, style-variant paraphrase pairs are
obtained by manual conversion to informal sentences in consideration of
content preservation, and is to be released publicly as the first open text
style transfer dataset in Korean. Our contributions are the following:

  * •

We present a corpus construction scheme capable of performing multiple tasks
while enabling parallel sentence style transfer.

  * •

We release a Korean corpus where sentence formality style is well defined,
regarding the daily used questions and
commands111https://github.com/cynthia/stylekqc.

##  2 Related Work

In general, sentence style222In this paper, we view ‘formality’ in Korean as a
style, while interchangeably using ‘conversion’ and ‘transfer’. is handled
regarding tone and manner in writing, though with a subtle difference Brooks
(2020). However, previous researches about content-preserving style transfer
Logeswaran et al. (2018); Tian et al. (2018) do not seem to be only on tone in
that the change in sentiment may influence the core speaker intent.
Furthermore, most approaches were from the perspective of unsupervised
learning dos Santos et al. (2018); Bao et al. (2019), with less explored
fields of parallel style-variant corpus for supervised learning, which might
provide robust guidance for the generative pre-trained models nowadays Radford
et al. (2019).

This trend was similarly revealed in previous studies on Korean. Since the
early approaches follow the studies in English and other languages, sentiment
or stance-based style transfer have prevalently been suggested Lee et al.
(2019); Choi and Na (2019)333Most of the work are not in an internationally
readable format; thus, we note here the methods used in the papers.. In Hong
et al. (2018), the transfer regarding politeness suffix of the sentence enders
was considered at the same time maintaining the sentence meaning, mainly
regarding ‘hay-yo’ and ‘hap-syo’ enders which differ in the degree of
formality. However, it dealt only with the syntactic change, not the
modification in the lexicon, adverbs, or tone and manner of the speech, which
are all considered influential for the honorific system Strauss and Eun
(2005). In this regard, we thought that formality style transfer should be
well defined along with content preservation. Furthermore, there is no open
dataset for Korean style transfer that can be utilized for research and
commercial purposes. We aim to resolve the above issues by proposing a
straightforward and effective building scheme.

##  3 Proposed Scheme

We construct a corpus of Korean directives, namely questions and commands,
where the question consists of an alternative question (Alt. Q) or wh-question
(wh- Q), and the command consists of prohibition (PH) and requirement (REQ),
following Cho et al. (2020). In other words, we target four types of speech
acts and assume sentences that can be uttered to humans or artificial
intelligent (AI) agents. There are six topics involved in this: messenger,
calendar, weather and news, smart home, shopping, and entertainment, which
come from a recent survey on customers usage Lee et al. (2020). 12 workers
from different backgrounds were recruited. We required specifying two likes
and one dislike on the topic, and these preferences were taken into account
when creating a total of 6 subgroups with two people each.

We created a construction scheme that goes through the following three steps
to check its reliability while generating utterances of 5,000 per topic and
7,500 per speech act.

  1. 1.

Writing natural language queries

  2. 2.

Rewriting paraphrased queries in formal tone

  3. 3.

Converting the formal sentences to informal

#### Query generation

First, query generation is a process in which participants directly suggest
the core content of directives which are to be rewritten in a formal style. In
this process, participants were asked to write a natural language query for
each of the given two speech acts on the assigned topic444These were given by
the process managers in Cho et al. (2020), but here we let them be created by
the workers to make the contents more diverse and to benefit from the
preferences. Since the query structure differs by speech act type as in Cho et
al. (2020), the created queries did not overlap across the workers. The
queries were checked for suitability, to avoid personally identifiable stuff
or the ones that can cause social harm. A total of 125 queries were generated
for each (topic, act) pair. The example of queries per some (topic, act) is
shown below. All the queries are generated in Korean, but described here in
English for demonstrative purpose.

  * •

(Shopping, Alt. Q) The one that has better A/S between Samsung and Apple

  * •

(Entertainment, Wh-Q) The TV channel number where the news is on at 8:00 p.m.

  * •

(Messenger, PH) Not to turn on WeChat automatic update

  * •

(Smart home, REQ) To recharge the wireless vacuum cleaner in the multi-room

No particular style was considered in generation, but the workers were asked
to make up the expressions that fit with colloquial context and daily life.
Also, knowledge-intensive questions or queries with multiple contents were
asked for modification.

Figure 1: An example of query generation-formal sentence writing-informal
transferring, along with the gloss and translation (PRT particle, NMN
nominalizer, ACC accusative, FUT futuristic, DEC declarative, POL polite, IMP
imperative). Though not reflected in the English translation, the transferring
preserves the overall structure of the formal sentence as well as the core
content.

#### Writing formal paraphrases

Next is a process in which the workers of the subgroups exchange queries
generated by each other and rewrite them into formal style sentences555In this
process, the workers check the validity of the query created by each other,
that the incompleteness of the queries that the moderator could have omitted
can be pointed out.. We primarily asked for the formal style because there are
more diverse expressions for formal utterances in the Korean language
regarding indirect speech and honorifics Byon (2006), so that it is easier for
paraphrasing compared to informal ones that might not come to the worker’s
mind at the first place. It was required that the utterances fit with the
conversation with the senior or elderly addressees rather than the friends or
juniors.

Rewriting was required for a total of 5 sentences. To make the paraphrases as
diverse as possible, the asking strategies in Byon (2006) and Cho (2008) were
requested. We display some excerpts:

  * •

Softening the commands to requests

  * •

Mentioning the addressee’s responsibility

  * •

Lessening the addressee’s burden

  * •

Asking the availability of the addressee

Some of these characteristics are shared across the culture Brown et al.
(1987). It may also be exhibited similar in the East Asian society Gu (1990)
and within similar syntax such as Japanese Okamoto (1999); Fukada and Asato
(2004). However, we faced language-specific considerations regarding the
functional and lexical expressions and asked the workers to reflect them in
the construction. Simultaneously, to fit with the naturalness within
colloquial context, written-style or outdated phrases/words were eschewed.

#### Converting to informal style

The final process is modifying directive sentences written in formal style
into informal sentences. Here, the workers convert the other person’s formal
sentences, created from the original query they had generated, checking the
typos and misunderstandings once again. ‘Informality’ defined here is slightly
different from being rude or impolite, but instead means the conversation
moves towards a more comfortable and personal relationship. Rao and Tetreault
(2018).

In this process, we asked the workers to maintain the overall sentence
structure, of which the diversity was already obtained owing to policies in
writing formal sentences. With this, we could prevent the potential overlap
between the converted sentences and also guarantee ‘parallel’ data. This can
be more effective in the Korean language where the indirectness is often
distinguished from formality; for instance, a cautious request to a younger
brother can be informal but indirect.

Style conversion was performed in various aspects such as change in sentence
enders, honorifics, and lexicons (such as nation to country). The workers were
encouraged to insert or delete some phrases depending on the naturalness
regarding the content, and to perform at least two word-level modifications.
The detailed
guideline666https://docs.google.com/document/d/1gjyEMCcp0mxmdzSKdd5OrLFVikyq22OsxXHisSr2THY
for the whole process was provided to the workers with the example query-
sentence tuples, and we exhibit one of them (Figure 1).

#### Refinement

The corpus was refined by three native speakers with corpus construction
experience for Korean directive sentences. In this process, typos, awkward
sentences, and paraphrases that are not sufficiently diverse were inspected,
and the reviews were reflected by the moderators.

##  4 Experiment

###  4.1 Task Setting

Through the experiment, we display that the proposed construction scheme
provides a corpus that can simultaneously enable multiple tasks, which can
bring advantage from a practical viewpoint.

  * •

Topic classification

  * •

Speech act classification

  * •

Paraphrase detection

  * •

Sentence style transfer

###  4.2 Implementation

For each of the total 24 [topic, act] chunks where we have 125 queries each,
we set aside 80% (100 queries) for training, 4% (5 queries) for validation,
and 16% (20 queries) for the test. From the whole dataset of volume 30,000,
the training set contains 24,000 sentences and 1,200/4,800 for dev/test each.
The queries were chosen randomly, and all the sets have an equal rate of topic
and speech act ratio.

Topic (TOPIC) and speech act (ACT) classification are intuitively formulated.
There are 5,000 utterances for each topic and 7,500 utterances for each speech
act, where six topics and four speech acts are set for the labels.

Paraphrase detection (PARA) requires a sentence pair. In Cho et al. (2020),
the sentence similarity was defined in 5-fold, checking if topic or speech act
overlaps between the two input sentences, with the highest similarity if the
queries are identical (the paraphrases). The paraphrase detection task was
derived from here by formulating the multi-class problem into a binary task.

Finally, we checked whether sentence style transfer (STYLE) works using the
pairs within; 12,000 pairs for training, 600 for validation and 2,400 for the
test. The training was done in the way of converting the formal sentences to
informal one.

Both sentence classification and paraphrase detection tasks were implemented
based on a BERT-based Devlin et al. (2019)
KcBERT777https://github.com/Beomi/KcBERT Lee (2020), and for sentence style
transfer, KoGPT2888https://github.com/SKT-AI/KoGPT2 that bases on GPT2 Radford
et al. (2019) was adopted. F1 (macro) and accuracy were used for the
classification tasks, and for style transfer, we checked character edit
distance (CED). The accuracy for style transfer ($\dagger$) denotes the
precision obtained with the model learned upon the train set Pang (2019).
Experimental settings are provided as supplementary.

| TOPIC | ACT | PARA | STYLE
---|---|---|---|---
Input | Sentence | Sentence | Pair | Sentence
Class # | 6 | 4 | 2 | -
Volume | 30,000 | 30,000 | 270,000 | 15,000
F1 Score | 92.68 | 97.75 | 99.93 | -
Accuracy | 92.83 | 97.75 | 99.93 | 99.58$\dagger$
CED | - | - | - | 0.451

Table 1: Experiment results on four subtasks.

###  4.3 Results

In classification and inference, we have the evaluation results that show
consistency between the train and test dataset (Table 1). Considering that the
queries in each set are distinguished from each other, we claim that our
dataset displays the extensibility to wider world problems, also given the
comprehensive coverage of topics and acts that are of interest in usual
conversation and smart speaker dialogues. Though the baseline score is quite
high for ACT and PARA, it does not harm one of our goals to provide a solid
scheme for corpus construction that suffices practical, real-world
applicability.

On STYLE, we adopted CED since formality as our ‘style’ more regards the
change in suffix and some lexicons rather than the whole word order and phrase
usage999On using other objective measures, the morpheme-level tokenization is
not yet unified for Korean sentences, to make evaluation harder.. Nonetheless,
we found the transfer task still challenging in view of the objective measure.
Instead, we observed the practical validity using the style classifier learned
upon train and valid set. We qualitatively checked that the seq2seq Sutskever
et al. (2014) with pre-trained generative model guarantees the intended style
transfer, and the detail is to be provided in Appendix A.

###  4.4 Discussion

We have some notes on the validity of the created dataset. Primarily, though
the dataset is first suggested open corpus for Korean style transfer, the
granularity of the style difference within the pair is not provided here as in
Rao and Tetreault (2018). Also, since our dataset provides the style transfer
that maintains the overall sentence structure, some sentence pairs show minor
differences, which is sufficient for spoken language processing but less
robust to digitized online texts. Finally, since the formality conversion
regards morpho-syntactic and lexical changes rather than the paraphrasing done
in writing the formal sentences, the diversity of expression regarding the
style is limited to the sentence formats that are not awkward to utter.

Despite the limitations, we want to emphasize that our approach can suggest a
reliable and efficient scheme for the service providers or task managers
aiming at a particular style transfer for various types of sentences. For
instance, if one replaces the queries with some structured query language
(SQL) or canonical forms of statements and use ‘rudeness’ or ‘twitter-
likeness’ as a style, the parallel dataset can be created in the same way,
though with a slightly different guideline. This kind of pair generation was
done with rule or back translation in Rao and Tetreault (2018), but we believe
that human-aided construction is more reliable and the resulting shortage of
data can be covered with the pre-trained models for the spoken language.

##  5 Conclusion

In this paper, we construct and disclose the first style-variant Korean
paraphrase corpus. Topic, speech act, and paraphrase are simultaneously
considered in evaluating the final corpus, where the consistent composition is
assumed to be guaranteed by the evaluation results. The entire guideline is
currently specific to the formality transfer in Korean, but can be utilized in
making up other parallel style transfer corpus with an extended pool of
topics, speech acts, queries, and style. All the resources are available
online101010https://github.com/cynthia/stylekqc.

## Ethical Considerations

In the corpus construction procedure which bases upon the documented approval
of the workers, adequate compensation was paid to each of them, in all the
process of query generation, writing formal sentences, and transferring them
to the informal one. The participants, recruited from the social media and
web, were familiar with the smart speakers and some of them had experience in
corpus construction processes. For 12 participants, 250 WON ($\approx$$0.22)
was provided in writing each query and 200 WON ($\approx$$0.18) for making up
the sentences. Thus, each participant was paid 600,000 WON ($\approx$$540) to
make up 250 queries and write 2,500 sentences.

Our resource is free from the license issue since all the materials were
created according to the guideline (a kind of template) and checked for post-
processing. The outcome of our project does not contain any personally
identifiable information, nor the contents that can induce social harm.

## Acknowledgements

This research was supported by the Technology Innovation Program (10076583,
Development of free-running speech recognition technologies for embedded robot
system) funded By the Ministry of Trade, Industry & Energy (MOTIE, Korea).
Also, the corpus construction was possible thanks to the help of twelve
passionate participants, namely Kyung Seo Ki, Dongho Lee, Yoon Kyung Lee, Hee
Young Park, Yulhee Kim, Seyoung Park, Jiwon An, Jeonghwa Cho, Kihyo Park,
Kyuhwan Lee, Soomin Lee, and Minhwa Chung.

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change of integration variables in (5.4) from $t$ to $s$ via

$t=\left(\frac{n+1}{n}\zeta\right)^{1/(n+1)}s.$

This leads us to the representation

$y_{0}(\zeta)=\sqrt{\frac{n}{2\pi}}\left(\frac{n+1}{n}\zeta\right)^{1/(n+1)}\mathrm{e}^{\zeta}\int_{\infty\mathrm{e}^{-\pi\mathrm{i}/(n+1)}}^{\infty\mathrm{e}^{\pi\mathrm{i}/(n+1)}}\exp\left(-\zeta\left(\frac{s^{n+1}}{n}-\frac{n+1}{n}s+1\right)\right)\mathrm{d}s.$

Utilising Cauchy’s theorem, we deform the integration contour to traverse
through the saddle point at $s=1$. Thus,

$y_{0}(\zeta)=\sqrt{\frac{n}{2\pi}}\left(\frac{n+1}{n}\zeta\right)^{1/(n+1)}\mathrm{e}^{\zeta}\int_{\mathscr{C}(\arg\zeta)}\exp\left(-\zeta\left(\frac{s^{n+1}}{n}-\frac{n+1}{n}s+1\right)\right)\mathrm{d}s,$

where $\mathscr{C}(\arg\zeta)$ denotes the path of steepest descent passing
through the saddle point at $s=1$. Varying the phase of $\zeta$ results in an
analytic continuation of $y_{0}(\zeta)$. This continuation remains valid as
long as the path $\mathscr{C}(\arg\zeta)$ varies continuously with respect to
$\arg\zeta$. When $n=3$, Figure 3 illustrates the path
$\mathscr{C}(\arg\zeta)$ for different values of $\arg\zeta$. When
$\arg\zeta=\frac{n-2}{2n}\pi$, the path connects to the saddle point
$\mathrm{e}^{2\pi\mathrm{i}/n}$. Likewise, when
$\arg\zeta=\frac{3n+2}{2n}\pi$, the path connects to the saddle point
$\mathrm{e}^{2\pi\mathrm{i}(n-1)/n}$. Consequently,
$\arg\zeta=\frac{n-2}{2n}\pi$ and $\arg\zeta=\frac{3n+2}{2n}\pi$ are Stokes
lines for $y_{0}(\zeta)$. An application of the method of steepest descents
then yields the asymptotic expansion

(5.5)
$\displaystyle\begin{split}y_{0}(\zeta)&\sim\left(\frac{n+1}{n}\zeta\right)^{1/(n+1)-1/2}\mathrm{e}^{\zeta}\sum_{m=0}^{\infty}\frac{a_{m}}{\zeta^{m}}=x^{(1/n-1)/2}\mathrm{e}^{\zeta}\sum_{m=0}^{\infty}\frac{a_{m}}{\zeta^{m}}\\\
&=x^{(1/n-1)/2}\mathrm{e}^{\zeta}\bigg{(}1+\frac{{2n^{2}-n-1}}{{24(n+1)}}\frac{1}{\zeta}+\frac{{4n^{4}+44n^{3}+21n^{2}-46n-23}}{{1152(n+1)^{2}}}\frac{1}{{\zeta^{2}}}+\ldots\bigg{)},\end{split}$

as $\zeta\to\infty$ in the sector
$-\frac{\pi}{n}+\delta\leq\arg\zeta\leq\frac{2n+1}{n}\pi-\delta$, with any
fixed $\delta>0$. The coefficients $a_{m}$ can be expressed using Perron’s
formula:

$a_{m}=\frac{1}{(2n+2)^{m}m!}\left[\frac{\mathrm{d}^{2m}}{\mathrm{d}s^{2m}}\left(\frac{n(n+1)}{2}\frac{(s-1)^{2}}{s^{n+1}-(n+1)s+n}\right)^{m+1/2}\right]_{s=1}$

(see, e.g., [5, Eq. 11]).

Figure 3. The steepest descent contour $\mathscr{C}(\arg\zeta)$ through the
saddle point $s=1$ when $n=3$ and (i) $\arg\zeta=\pi$, (ii)
$\arg\zeta=\frac{\pi}{2}$ and (iii) $\arg\zeta=\frac{\pi}{5}$. The bullet
points represent the saddles $1$, $\mathrm{e}^{2\pi\mathrm{i}/3}$ and
$\mathrm{e}^{4\pi\mathrm{i}/3}$.

We can readily verify that the function $w(u,z)=y(u^{n/(n+1)}z)$ satisfies
equation (5.1). Consequently,

$W_{0}(u,\xi)=z^{(1-1/n)/2}w(u,z)=u^{1/2-1/(n+1)}z^{(1-1/n)/2}y_{0}(u\xi)$

coincides with the solution (5.3) obtained from Theorem 1.1. From (5.5),

(5.6)
$W_{0}(u,\xi)\sim\mathrm{e}^{u\xi}\sum\limits_{m=0}^{\infty}\frac{a_{m}}{(u\xi)^{m}},$

as $u\xi\to\infty$ within the sector
$-\frac{\pi}{n}+\delta\leq\arg(u\xi)\leq\frac{2n+1}{n}\pi-\delta$, for any
fixed $\delta>0$. In particular, this implies
$\mathsf{A}_{0,m}(\xi)=a_{m}\xi^{-m}$. Therefore, for the asymptotic expansion
(5.6) to remain valid as $u\to\infty$ within any half-plane
$\operatorname{Re}(u)\geq\sigma>0$, $\xi$ must be confined within a sector
$\frac{n-2}{2n}\pi+\delta\leq\arg\xi\leq\frac{3n+2}{2n}\pi-\delta$, with any
fixed $\delta>0$, and $|\xi|$ must be bounded away from $0$. Such a set is
properly contained in a $\Gamma_{0}(d)$, resulting from Theorem 1.1 with a
suitably chosen $d>0$.

##  6\. Discussion

We studied the Borel summability of formal solutions of certain
$n^{\text{th}}$-order linear ordinary differential equations with a large
parameter. We demonstrated that, given mild conditions on the potential
functions of the equation, the formal solutions are Borel summable with
respect to the large parameter in vast, unbounded domains of the independent
variable. We established that the formal series expansions serve as asymptotic
expansions, uniform with respect to the independent variable, for the Borel
re-summed exact solutions. Additionally, we showed that the exact solutions
can be expressed using factorial series in the parameter, and these expansions
converge in half-planes, uniformly in the independent variable.

The theory presented here marks an initial step toward a global analysis of
formal solutions of the differential equation (1.1). The primary result of
this paper demonstrates the Borel summability of formal solutions within a
Stokes region, provided that the independent variable remains bounded away
from Stokes curves emerging either from a zero of $f_{0,0}(z)$ or from a
singularity of one of the functions $f_{k}(u,z)$. Importantly, it does not
provide any information regarding connection formulae joining formal solutions
across the Stokes curves.

In the following discussion, we briefly explore a potential extension of the
theory to turning point problems. Turning points represent the simplest form
of transition points. For illustration, consider the following third-order
differential equation:

(6.1)
$\bigg{(}-\frac{\mathrm{d}^{3}}{\mathrm{d}z^{3}}+u^{3}f(z)\bigg{)}w(u,z)=0.$

Suppose that the potential function $f(z)$ is analytic in a domain
$\mathbf{D}$ and $f(z)$ vanishes at exactly one point $z_{0}$, say, of
$\mathbf{D}$. For the sake of simplicity, we assume that $z_{0}$ is a simple
zero of $f(z)$. It is convenient to work in terms of the transformed variables
$\zeta$ and $\mathcal{W}(u,\zeta)$ given by

(6.2)
$\frac{3}{4}\zeta^{4/3}=\int_{z_{0}}^{z}f^{1/3}(t)\mathrm{d}t,\quad\mathcal{W}(u,\zeta)=\zeta^{-1/3}f^{1/3}(z)w(u,z).$

Subsequently, equation (6.1) transforms into

(6.3)
$\bigg{(}-\frac{\mathrm{d}^{3}}{\mathrm{d}\zeta^{3}}+\phi(\zeta)\frac{\mathrm{d}}{\mathrm{d}\zeta}+u^{3}\zeta+\psi(\zeta)\bigg{)}\mathcal{W}(u,\zeta)=0,$

with

$\phi(\zeta)=\bigg{(}\frac{\mathrm{d}\widehat{f}^{-1/3}(z)}{{\mathrm{d}z}}\bigg{)}^{2}-2\widehat{f}^{-1/3}(z)\frac{\mathrm{d}^{2}\widehat{f}^{-1/3}(z)}{\mathrm{d}z^{2}},\quad\psi(\zeta)=-\widehat{f}^{-2/3}(z)\frac{{\mathrm{d}^{3}\widehat{f}^{-1/3}(z)}}{{\mathrm{d}z^{3}}}$

and $\widehat{f}(z)=\zeta^{-1}f(z)$. The functions $\phi(\zeta)$ and
$\psi(\zeta)$ are analytic in the corresponding $\zeta$ domain $\mathbf{H}$,
say. The transformation (6.2) maps the simple turning point $z_{0}$ into the
origin in $\mathbf{H}$, and the four Stokes curves issuing from $z_{0}$ are
mapped into the rays
$\arg\zeta=0,\frac{\pi}{2}\mathrm{i},\pi\mathrm{i},\frac{3\pi}{2}\mathrm{i}$.
Let $Y(x)$ be any solution of the third-order Airy-type equation

(6.4) $\bigg{(}-\frac{\mathrm{d}^{3}}{\mathrm{d}x^{3}}+x\bigg{)}Y(x)=0.$

It can be verified by direct substitution that equation (6.3) has a formal
solution of the form

(6.5)
$\mathcal{W}(u,\zeta)=Y(u^{3/4}\zeta)\sum\limits_{m=0}^{\infty}\frac{A_{m}(\zeta)}{u^{3m}}+u^{-9/4}Y^{\prime}(u^{3/4}\zeta)\sum\limits_{m=0}^{\infty}\frac{B_{m}(\zeta)}{u^{3m}}+u^{-3/2}Y^{\prime\prime}(u^{3/4}\zeta)\sum\limits_{m=0}^{\infty}\frac{C_{m}(\zeta)}{u^{3m}}.$

The coefficients $A_{m}(\zeta)$, $B_{m}(\zeta)$ and $C_{m}(\zeta)$, which are
analytic functions of $\zeta$ in $\mathbf{H}$, can be determined recursively
via

$\displaystyle A_{0}(\zeta)$ $\displaystyle=1,\quad
C_{0}(\zeta)=\frac{1}{3}\zeta^{-2/3}\int_{0}^{\zeta}t^{-1/3}\phi(t)\mathrm{d}t,$
$\displaystyle B_{0}(\zeta)$
$\displaystyle=-\frac{1}{3}\zeta^{-1/3}\int_{0}^{\zeta}t^{-2/3}\left(3(tC^{\prime}_{0}(t))^{\prime}-\phi(t)tC_{0}(t)-\psi(t)\right)\mathrm{d}t,$

and

$\displaystyle A_{m}(\zeta)=$
$\displaystyle-\frac{1}{3}\int^{\zeta}\left(3B^{\prime\prime}_{m-1}(t)-\phi(t)B_{m-1}(t)+C^{\prime\prime\prime}_{m-1}(t)-\phi(t)C^{\prime}_{m-1}(t)-\psi(t)C_{m-1}(t)\right)\mathrm{d}t,$
$\displaystyle C_{m}(\zeta)=$
$\displaystyle-\frac{1}{3}\zeta^{-2/3}\int_{0}^{\zeta}t^{-1/3}\left(3A^{\prime\prime}_{m}(t)-\phi(t)A_{m}(t)+B^{\prime\prime\prime}_{m-1}(t)-\phi(t)B^{\prime}_{m-1}(t)-\psi(t)B_{m-1}(t)\right)\mathrm{d}t,$
$\displaystyle B_{m}(\zeta)=$
$\displaystyle-\frac{1}{3}\zeta^{-1/3}\int_{0}^{\zeta}t^{-2/3}\left(A^{\prime\prime\prime}_{m}(t)-\phi(t)A^{\prime}_{m}(t)-\psi(t)A_{m}(t)+3(tC^{\prime}_{m}(t))^{\prime}-\phi(t)tC_{m}(t)\right)\mathrm{d}t,$

for $m\geq 1$. The constants of integration in the expression for
$A_{m}(\zeta)$ can be chosen arbitrarily. We anticipate that, assuming
appropriate conditions on the functions $\phi(\zeta)$ and $\psi(\zeta)$, and
for sufficiently large values of $\operatorname{Re}(u)$, the three formal
series in (6.5) are Borel-summable within a specific, well-defined subdomain
of $\mathbf{H}$ that includes $\zeta=0$ and the associated Stokes rays
$\arg\zeta=0,\frac{\pi}{2}\mathrm{i},\pi\mathrm{i},\frac{3\pi}{2}\mathrm{i}$.
This could potentially be confirmed by employing an argument similar to that
used in the second-order case [17]. As a result, the continuation formulae for
formal solutions of the form (6.5) can be derived directly from those
corresponding to the solutions of (6.4) (see, e.g., [31]). Subsequently, by
employing the transformation

$\xi=\frac{3}{4}\zeta^{4/3},\quad W(u,\xi)=\zeta^{1/3}\mathcal{W}(u,\zeta),$

connection formulae can be established for formal solutions of the form (1.7).

Finally, extending the results of the paper to differential equations of the
type (1.1), where the terms corresponding to $m=0$ in the expansions (1.2) of
the functions $f_{k}(u,z)$ (for $k\neq 0$) need not be zero, would be of great
interest. We anticipate that the emergence of potential new Stokes curves
could pose challenges to the analysis.

## Acknowledgement

This research was conducted in part during a visit to the Okinawa Institute of
Science and Technology (OIST) as part of the Theoretical Sciences Visiting
Program (TSVP). The author’s research was supported by the JSPS Postdoctoral
Research Fellowship No. P21020 and by the JSPS KAKENHI Grant No. JP21F21020.

##  Appendix A The transformation of the equation

In this appendix, we derive expressions that relate the functions
$\psi_{k}(u,\xi)$ and $\psi_{k,m}(\xi)$ to their respective counterparts
$f_{k}(u,z)$ and $f_{k,m}(z)$. The basis for the derivation is the following
lemma.

######  Lemma A.1.

For any $p\geq 1$, the following identity between differential operators
holds:

(A.1)
$\frac{\mathrm{d}^{p}}{\mathrm{d}z^{p}}=\sum\limits_{r=1}^{p}\text{\large$\mathtt{B}$}_{p,r}\bigg{(}f_{0,0}^{1/n}(z),\frac{\mathrm{d}f_{0,0}^{1/n}(z)}{\mathrm{d}z},\ldots,\frac{\mathrm{d}^{p-r}f_{0,0}^{1/n}(z)}{\mathrm{d}z^{p-r}}\bigg{)}\frac{\mathrm{d}^{r}}{\mathrm{d}\xi^{r}}.$

###### Proof.

We proceed by induction on $p$. The base case, when $p=1$, is established
through the chain rule and the equality

$\frac{\mathrm{d}\xi}{\mathrm{d}z}=f_{0,0}^{1/n}(z).$

This equality is a direct consequence of (1.4). Suppose the identity (A.1)
holds for a given positive integer $p$. Using both the induction hypothesis
and the base case, and applying the identity (C.3) and the recurrence relation
(C.1), we deduce

$\displaystyle\frac{\mathrm{d}^{p+1}}{\mathrm{d}z^{p+1}}$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}z}\frac{\mathrm{d}^{p}}{\mathrm{d}z^{p}}=\sum\limits_{r=1}^{p}\frac{\mathrm{d}\text{\large$\mathtt{B}$}_{p,r}}{{\mathrm{d}z}}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}+\sum\limits_{r=1}^{p}{f_{0,0}^{1/n}(z)\text{\large$\mathtt{B}$}_{p,r}\frac{{\mathrm{d}^{r+1}}}{{\mathrm{d}\xi^{r+1}}}}$
$\displaystyle=\sum\limits_{r=1}^{p}{\bigg{(}{\sum\limits_{q=1}^{p-r+1}{\binom{p}{q}\frac{{\mathrm{d}^{q}f_{0,0}^{1/n}(z)}}{{\mathrm{d}z^{q}}}\text{\large$\mathtt{B}$}_{p-q,r-1}}}\bigg{)}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}+\sum\limits_{r=1}^{p+1}{f_{0,0}^{1/n}(z)\text{\large$\mathtt{B}$}_{p,r-1}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}$
$\displaystyle=f_{0,0}^{1/n}(z)\text{\large$\mathtt{B}$}_{p,p}\frac{{\mathrm{d}^{p+1}}}{{\mathrm{d}\xi^{p+1}}}+\sum\limits_{r=1}^{p}{\bigg{(}f_{0,0}^{1/n}(z)\text{\large$\mathtt{B}$}_{p,r-1}+\sum\limits_{q=1}^{p-r+1}{\binom{p}{q}\frac{{\mathrm{d}^{q}f_{0,0}^{1/n}(z)}}{{\mathrm{d}z^{q}}}\text{\large$\mathtt{B}$}_{p-q,r-1}}\bigg{)}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}$
$\displaystyle=\text{\large$\mathtt{B}$}_{p+1,p+1}\frac{{\mathrm{d}^{p+1}}}{{\mathrm{d}\xi^{p+1}}}+\sum\limits_{r=1}^{p}{\bigg{(}\sum\limits_{q=0}^{p-r+1}{\binom{p}{q}\frac{{\mathrm{d}^{q}f_{0,0}^{1/n}(z)}}{{\mathrm{d}z^{q}}}\text{\large$\mathtt{B}$}_{p-q,r-1}}\bigg{)}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}$
$\displaystyle=\text{\large$\mathtt{B}$}_{p+1,p+1}\frac{{\mathrm{d}^{p+1}}}{{\mathrm{d}\xi^{p+1}}}+\sum\limits_{r=1}^{p}{\bigg{(}\sum\limits_{q=1}^{p+1-r+1}{\binom{p+1-1}{q-1}\frac{{\mathrm{d}^{q-1}f_{0,0}^{1/n}(z)}}{{\mathrm{d}z^{q-1}}}\text{\large$\mathtt{B}$}_{p+1-q,r-1}}\bigg{)}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}$
$\displaystyle=\text{\large$\mathtt{B}$}_{p+1,p+1}\frac{{\mathrm{d}^{p+1}}}{{\mathrm{d}\xi^{p+1}}}+\sum\limits_{r=1}^{p}{\text{\large$\mathtt{B}$}_{p+1,r}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}=\sum\limits_{r=1}^{p+1}{\text{\large$\mathtt{B}$}_{p+1,r}\frac{{\mathrm{d}^{r}}}{{\mathrm{d}\xi^{r}}}}.$

This concludes the induction step. ∎

Upon differentiating the equality $w(u,z)=f_{0,0}^{(1/n-1)/2}(z)W(u,\xi)$ $k$
times with respect to $z$ and applying Lemma A.1, we obtain

$\displaystyle\frac{\mathrm{d}^{k}w(u,z)}{\mathrm{d}z^{k}}$
$\displaystyle=\sum\limits_{p=0}^{k}\binom{k}{p}\frac{\mathrm{d}^{k-p}f_{0,0}^{(1/n-1)/2}(z)}{\mathrm{d}z^{k-p}}\frac{\mathrm{d}^{p}W(u,\xi)}{\mathrm{d}z^{p}}$
$\displaystyle=\frac{\mathrm{d}^{k}f_{0,0}^{(1/n-1)/2}(z)}{\mathrm{d}z^{k}}W(u,\xi)+\sum\limits_{p=1}^{k}\binom{k}{p}\frac{\mathrm{d}^{k-p}f_{0,0}^{(1/n-1)/2}(z)}{\mathrm{d}z^{k-p}}\sum_{r=1}^{p}\text{\large$\mathtt{B}$}_{p,r}\frac{\mathrm{d}^{p}W(u,\xi)}{\mathrm{d}\xi^{p}}.$

By substituting into (1.1) and re-arranging the expression, we derive the
equation (1.5) with

(A.2)
$\displaystyle\begin{split}\psi_{k}(u,\xi)=&-\frac{{u^{k-n}}}{{f_{0,0}^{(1+1/n)/2}(z)}}\sum\limits_{p=k}^{n}{\binom{n}{p}\frac{{\mathrm{d}^{n-p}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{n-p}}}}\text{\large$\mathtt{B}$}_{p,k}\\\
&+\frac{1}{{f_{0,0}^{(1+1/n)/2}(z)}}\sum\limits_{p=0}^{n-2}{u^{k-p}f_{p}(u,z)\sum\limits_{r=k}^{p}{\binom{p}{r}\frac{{\mathrm{d}^{p-r}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{p-r}}}\text{\large$\mathtt{B}$}_{r,k}}}.\end{split}$

Substituting the expansions (1.2) into this expression and re-arranging the
resulting equation in descending powers of $u$, we derive the following
formulae for the coefficients $\psi_{k,m}(\xi)$. For $k=0$, we obtain

$\psi_{0,1}(\xi)=\frac{{f_{0,1}(z)}}{{f_{0,0}(z)}},$
$\psi_{0,n}(\xi)=\frac{{f_{0,n}(z)}}{{f_{0,0}(z)}}-\frac{1}{f_{0,0}^{(1+1/n)/2}(z)}\bigg{(}\frac{{\mathrm{d}^{n}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{n}}}-\sum\limits_{p=1}^{n-2}{f_{p,n-p}(z)\frac{{\mathrm{d}^{p}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{p}}}}\bigg{)},$

and

$\psi_{0,m}(\xi)=\frac{{f_{0,m}(z)}}{{f_{0,0}(z)}}+\frac{1}{f_{0,0}^{(1+1/n)/2}(z)}\sum\limits_{p=1}^{\min(n-2,m-1)}{f_{p,m-p}(z)\frac{{\mathrm{d}^{p}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{p}}}}$

for $n\neq m\geq 2$. For $1\leq k\leq n-2$, we find

$\displaystyle\psi_{k,n-k}(\xi)=\;$
$\displaystyle\frac{1}{f_{0,0}^{(1+1/n)/2}(z)}\sum\limits_{p=0}^{n-k-2}{f_{k+p,n-k-p}(z)\sum\limits_{r=0}^{p}{\binom{k+p}{k+r}\frac{{\mathrm{d}^{p-r}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{p-r}}}}\text{\large$\mathtt{B}$}_{k+r,k}}$
$\displaystyle-\frac{1}{f_{0,0}^{(1+1/n)/2}(z)}\sum\limits_{r=k}^{n}{\binom{n}{r}\frac{{\mathrm{d}^{n-r}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{n-r}}}\text{\large$\mathtt{B}$}_{r,k}},$

and

$\psi_{k,m}(\xi)=\frac{1}{f_{0,0}^{(1+1/n)/2}(z)}\sum\limits_{p=0}^{\min(n-k-2,m-1)}{f_{k+p,m-p}(z)\sum\limits_{r=0}^{p}{\binom{k+p}{k+r}\frac{{\mathrm{d}^{p-r}f_{0,0}^{(1/n-1)/2}(z)}}{{\mathrm{d}z^{p-r}}}}\text{\large$\mathtt{B}$}_{k+r,k}}$

for $n-k\neq m\geq 1$.

##  Appendix B Recurrence relation for the coefficients

In this appendix, we outline the derivation of the recurrence relation (1.9)
satisfied by the coefficients $\mathsf{A}_{j,m}(\xi)$. Letting
$\mathsf{A}_{j,0}(\xi)=1$ and
$\text{\large$\mathtt{B}$}_{r}=\text{\large$\mathtt{B}$}_{r}(X^{\prime}_{j}(\xi),X^{\prime\prime}_{j}(\xi),\ldots,X_{j}^{(r)}(\xi))$,
we employ (C.5) to obtain the following expansions for the derivatives of the
formal solutions (1.7):

$\displaystyle\exp\Big{(}{-\mathrm{e}^{2\pi\mathrm{i}j/n}u\xi-
X_{j}(\xi)}\Big{)}\frac{{\mathrm{d}^{k}}}{{\mathrm{d}\xi^{k}}}\bigg{(}{\exp\Big{(}{\mathrm{e}^{2\pi\mathrm{i}j/n}u\xi+X_{j}(\xi)}\Big{)}\sum\limits_{m=0}^{\infty}{\frac{{\mathsf{A}_{j,m}(\xi)}}{{u^{m}}}}}\bigg{)}$
$\displaystyle=\sum\limits_{p=0}^{k}{\binom{k}{p}\bigg{(}{\sum\limits_{r=0}^{p}{\binom{p}{r}\mathrm{e}^{2\pi\mathrm{i}j(p-r)/n}u^{p-r}\text{\large$\mathtt{B}$}_{r}}}\bigg{)}\bigg{(}{\sum\limits_{m=0}^{\infty}{\frac{{\mathrm{d}^{k-p}\mathsf{A}_{j,m}(\xi)}}{{\mathrm{d}\xi^{k-p}}}\frac{1}{{u^{m}}}}}\bigg{)}}$
$\displaystyle=\sum\limits_{m=0}^{\infty}{\bigg{(}{\sum\limits_{p=0}^{k}{\mathrm{e}^{2\pi\mathrm{i}jp/n}\bigg{(}{\sum\limits_{r=p}^{k}{\binom{k}{r}\binom{r}{r-p}\text{\large$\mathtt{B}$}_{r-p}\frac{{\mathrm{d}^{k-r}\mathsf{A}_{j,m}(\xi)}}{{\mathrm{d}\xi^{k-r}}}}}\bigg{)}u^{p}}}\bigg{)}\frac{1}{{u^{m}}}}$
$\displaystyle=u^{k}\sum\limits_{m=0}^{\infty}{\sum\limits_{p=0}^{k}{\mathrm{e}^{2\pi\mathrm{i}j(k-p)/n}\sum\limits_{r=0}^{p}{\binom{k}{r}\binom{k-r}{p-r}\text{\large$\mathtt{B}$}_{p-r}\frac{{\mathrm{d}^{r}\mathsf{A}_{j,m}(\xi)}}{{\mathrm{d}\xi^{r}}}}\frac{1}{{u^{m+p}}}}}$
$\displaystyle=u^{k}\sum\limits_{m=0}^{\infty}{\bigg{(}{\sum\limits_{p=0}^{\min(m,k)}{\mathrm{e}^{2\pi\mathrm{i}j(k-p)/n}\sum\limits_{r=0}^{p}{\binom{k}{r}\binom{k-r}{p-r}\text{\large$\mathtt{B}$}_{p-r}\frac{{\mathrm{d}^{r}\mathsf{A}_{j,m-p}(\xi)}}{{\mathrm{d}\xi^{r}}}}}}\bigg{)}\frac{1}{{u^{m}}}}$
$\displaystyle=u^{k}\sum\limits_{m=0}^{\infty}{\bigg{(}{\sum\limits_{p=0}^{\min(m,k)}{\binom{k}{p}\mathrm{e}^{2\pi\mathrm{i}j(k-p)/n}\sum\limits_{r=0}^{p}{\binom{p}{r}\text{\large$\mathtt{B}$}_{p-r}\frac{{\mathrm{d}^{r}\mathsf{A}_{j,m-p}(\xi)}}{{\mathrm{d}\xi^{r}}}}}}\bigg{)}\frac{1}{{u^{m}}}}.$

Substituting both these expansions and those in (1.6) into equation (1.5),
performing the products of the expansions, and equating coefficients of like
powers of $u$, we obtain

$\sum\limits_{p=1}^{\min(n,m)}{\binom{n}{p}\mathrm{e}^{2\pi\mathrm{i}j(n-p)/n}\sum\limits_{r=0}^{p}{\binom{p}{r}\text{\large$\mathtt{B}$}_{p-r}\frac{{\mathrm{d}^{r}\mathsf{A}_{j,m-p}(\xi)}}{{\mathrm{d}\xi^{r}}}}}\\\
=\sum\limits_{k=0}^{n-2}{\sum\limits_{q=0}^{m-1}{\sum\limits_{p=0}^{\min(k,q)}{\binom{k}{p}\mathrm{e}^{2\pi\mathrm{i}j(k-p)/n}\sum\limits_{r=0}^{p}{\binom{p}{r}\text{\large$\mathtt{B}$}_{p-r}\psi_{k,m-q}(\xi)\frac{{\mathrm{d}^{r}\mathsf{A}_{j,q-p}(\xi)}}{{\mathrm{d}\xi^{r}}}}}}}.$

The terms involving $\mathsf{A}_{j,m}(\xi)$ cancel each other out. By solving
for $\partial_{\xi}\mathsf{A}_{j,m-1}(\xi)$, integrating both sides, and
substituting $m$ with $m+1$ throughout, we arrive at the desired recurrence
relation (1.9) for $m\geq 1$.

##  Appendix C Bell polynomials

Due to the frequent occurrences of the Bell polynomials in this paper, we have
compiled in this appendix their properties that are utilised throughout the
paper. The partial exponential Bell polynomials
$\text{\large$\mathtt{B}$}_{p,r}$ are a triangular array of polynomials
defined by the generating function

$\frac{1}{r!}\bigg{(}\sum\limits_{p=1}^{\infty}x_{p}\frac{z^{p}}{p!}\bigg{)}^{r}=\sum\limits_{p=r}^{\infty}\text{\large$\mathtt{B}$}_{p,r}(x_{1},x_{2},\ldots,x_{p-r+1})\frac{z^{p}}{p!},\quad
r\geq 0.$

They can be computed efficiently by a recurrence relation:

(C.1)
$\text{\large$\mathtt{B}$}_{p+1,r+1}(x_{1},x_{2},\ldots,x_{p-r+1})=\sum\limits_{q=0}^{p-r}\binom{p}{q}x_{q+1}\text{\large$\mathtt{B}$}_{p-q,r}(x_{1},x_{2},\ldots,x_{p-r-q+1}),$

where $\text{\large$\mathtt{B}$}_{0,0}=1$, $\text{\large$\mathtt{B}$}_{p,0}=0$
for $p\geq 1$, and $\text{\large$\mathtt{B}$}_{0,r}=0$ for $r\geq 1$. The
partial exponential Bell polynomials satisfy the following homogeneity
relation:

(C.2) $\text{\large$\mathtt{B}$}_{p,r}(\alpha\beta
x_{1},\alpha\beta^{2}x_{2},\ldots,\alpha\beta^{p-r+1}x_{p-r+1})=\alpha^{r}\beta^{p}\text{\large$\mathtt{B}$}_{p,r}(x_{1},x_{2},\ldots,x_{p-r+1}).$

By applying the chain rule, one can derive the following identity:

(C.3)
$\frac{\mathrm{d}}{\mathrm{d}z}\text{\large$\mathtt{B}$}_{p,r}(x_{1}(z),x_{2}(z),\ldots,x_{p-r+1}(z))=\sum_{q=1}^{p-r+1}\binom{p}{q}x_{q}^{\prime}(z)\text{\large$\mathtt{B}$}_{p-q,r-1}(x_{1}(z),x_{2}(z),\ldots,x_{p-q-r+2}(z)).$

The sum

$\text{\large$\mathtt{B}$}_{p}(x_{1},x_{2},\ldots,x_{p})=\sum_{r=0}^{p}\text{\large$\mathtt{B}$}_{p,r}(x_{1},x_{2},\ldots,x_{p-r+1})$

is called the $p^{\text{th}}$ complete exponential Bell polynomial. These
polynomials can be computed recursively via

(C.4)
$\text{\large$\mathtt{B}$}_{p+1}(x_{1},x_{2},\ldots,x_{p+1})=\sum\limits_{q=0}^{p}\binom{p}{q}x_{q+1}\text{\large$\mathtt{B}$}_{p-q}(x_{1},x_{2},\ldots,x_{p-q}),$

with the initial value $\text{\large$\mathtt{B}$}_{0}=1$. The exponential Bell
polynomials play a role in the celebrated Faà di Bruno formula, which
generalises the chain rule to higher derivatives. In particular,

(C.5)
$\frac{\mathrm{d}^{p}\exp(x(z))}{\mathrm{d}z^{p}}=\exp(x(z))\text{\large$\mathtt{B}$}_{p}(x^{\prime}(z),x^{\prime\prime}(z),\ldots,x^{(p)}(z)).$

For further properties of the Bell polynomials, we refer to [12, Ch. III].

##  Appendix D The sequence $\mathfrak{a}_{n}$

In this appendix, we establish certain properties of the sequence
$\mathfrak{a}_{n}$ with $n\geq 2$, where the $n^{\text{th}}$ term is defined
as the unique positive solution of the equation

$(1+\mathfrak{a}_{n})^{n}=1+2n\mathfrak{a}_{n}.$

The verification that this equation possesses a distinct positive solution for
each $n\geq 2$ is left to the reader.

######  Proposition D.1.

Let $a=1.2564312086\ldots$ be the unique positive solution of the equation
$\mathrm{e}^{a}=1+2a$. Then for any integer $n\geq 2$, the double inequality

(D.1) $\frac{a}{n}\leq\mathfrak{a}_{n}\leq\frac{2}{n-1}$

holds. Furthermore, the sequence $\mathfrak{a}_{n}$ satisfies the asymptotic
limit

(D.2) $\mathfrak{a}_{n}\sim\frac{a}{n},$

as $n\to+\infty$.

Here, we remark that the constant $a$ can be represented using particular
branches of the Lambert $W$-function (see, for instance, [34, §4.13]):

$a=-\tfrac{1}{2}-W_{\pm 1}\big{(}-\tfrac{1}{{2\sqrt{\mathrm{e}}}}\mp
0\mathrm{i}\big{)}.$

###### Proof.

First, we note that $1+\mathfrak{a}_{n}\leq\mathrm{e}^{\mathfrak{a}_{n}}$.
Consequently,

$1+2n\mathfrak{a}_{n}\leq\mathrm{e}^{n\mathfrak{a}_{n}},$

meaning

$2\leq\frac{{\mathrm{e}^{n\mathfrak{a}_{n}}-1}}{{n\mathfrak{a}_{n}}}.$

Since the function $x\mapsto(\mathrm{e}^{x}-1)/x$ is monotonically increasing
for $x>0$, the above inequality implies $a\leq n\mathfrak{a}_{n}$. On the
other hand, employing the binomial theorem yields

$(1+\mathfrak{a}_{n})^{n}\geq
1+n\mathfrak{a}_{n}+\frac{{n(n-1)}}{2}\mathfrak{a}_{n}^{2}$

for any $n\geq 2$. Consequently,

$1+n\mathfrak{a}_{n}+\frac{{n(n-1)}}{2}\mathfrak{a}_{n}^{2}\leq
1+2n\mathfrak{a}_{n}$

which can be simplified to

$\mathfrak{a}_{n}\leq\frac{2}{{n-1}}.$

To establish the asymptotic equality (D.2), we make note of the fact that
$(1+x)\mathrm{e}^{-x}=1+\mathcal{O}(x^{2})$ as $x\to 0$. Combining this with
the double inequality (D.1), we deduce that

$\lim\limits_{n\to+\infty}((1+\mathfrak{a}_{n})\mathrm{e}^{-\mathfrak{a}_{n}})^{n}=1.$

As the limit is positive and the exponential function exhibits monotonic
growth, we ascertain that

$\limsup\limits_{n\to+\infty}(1+\mathfrak{a}_{n})^{n}=\limsup\limits_{n\to+\infty}\mathrm{e}^{n\mathfrak{a}_{n}}\lim\limits_{n\to+\infty}((1+\mathfrak{a}_{n})\mathrm{e}^{-\mathfrak{a}_{n}})^{n}=\exp\Big{(}\limsup\limits_{n\to+\infty}(n\mathfrak{a}_{n})\Big{)}.$

Therefore,

$\exp\Big{(}\limsup\limits_{n\to+\infty}(n\mathfrak{a}_{n})\Big{)}\leq
1+2\limsup\limits_{n\to+\infty}(n\mathfrak{a}_{n}),$

which, given that $a\leq\limsup_{n\to+\infty}(n\mathfrak{a}_{n})\leq 2$ as
inferred from (D.1), implies

$\frac{\exp\Big{(}\limsup\limits_{n\to+\infty}(n\mathfrak{a}_{n})\Big{)}-1}{\limsup\limits_{n\to+\infty}(n\mathfrak{a}_{n})}\leq
2.$

Utilising once more the monotonous nature of the function
$x\mapsto(\mathrm{e}^{x}-1)/x$, we deduce

$\limsup\limits_{n\to+\infty}(n\mathfrak{a}_{n})\leq a,$

and with reference to (D.1), this results in the desired limit

$\lim\limits_{n\to+\infty}(n\mathfrak{a}_{n})=a.$

∎

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# Relaxation mechanism of GaP grown on 001 Si substrates: influence of defects
on the growth of AlGaP layers on GaP/Si templates.

K. Pantzas<EMAIL_ADDRESS>G. Beaudoin  M. Bailly  A.
Martin  A. Grisard  D. Dolfi  O. Mauguin  L. Largeau  I. Sagnes  G. Patriarche
Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies -
C2N, 91120, Palaiseau, France.  Thales Research & Technology, F-91120,
Palaiseau, France

###### Abstract

The mechanical stability of commercial GaP/Si templates during thermal
annealing and subsequent MOCVD growth of GaP and AlGaP is investigated.
Although the GaP layer of the template originally presents an excellent
surface morphology, annealing at high enough temperatures to remove the native
oxide prior to growth leads to plastic relaxation, accompanied by a variety of
defects, including a dense grid of micro-twins. These micro-twins
detrimentally affect GaP and AlGaP layers grown subsequently on the template.

##  1 Introduction

The (Al, Ga)P system possesses a significant number of properties highly
desirable for nanophotonics integrated on Si. A non-exhaustive list of these
properties includes: lattice mismatches of only
$0.36\text{\,}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$
between GaP and Si [1], the smallest among all III-V compounds and Si, and
$0.3\text{\,}\mathrm{\char 37\relax}$ between AlP and GaP, allowing for the
growth of thick, high-quality layers and Bragg mirrors [2, 3]; a refractive
index contrast as large as 0.4 at $700\text{\,}\mathrm{nm}$ and 0.57 at
$500\text{\,}\mathrm{nm}$. Only fifteen AlP/GaP pairs are, therefore, required
to yield a peak reflectivity of $98\text{\,}\mathrm{\char 37\relax}$ for
wavelengths between $530\text{\,}\mathrm{nm}690\text{\,}\mathrm{nm}$ [4];
band-gap energies of $2.5\text{\,}\mathrm{eV}$ and $2.24\text{\,}\mathrm{eV}$
for AlP and GaP, respectively, making both materials transparent to the green-
to-red region of the visible spectrum and allowing active structures to be
optically pumped by conventional green lasers; high second-order nonlinearity,
Pockels, and piezoelectric effects [3, 5]; high thermal conductivity,
$110\text{\,}\mathrm{W}\text{\,}{\mathrm{K}}^{-1}\text{\,}{\mathrm{m}}^{-1}$,
twenty times higher than that of active region material InGaP [6], and close
to that of Si.

Despite both the low lattice and thermal mismatch between Si and the (Al, Ga)P
system, their integration is not straightforward. Indeed, Si crystallizes in
the diamond, non-polar structure, while (Al,Ga)P materials crystallize in the
polar zinc-blend structure. This difference in polarity leads to the
apparition of anti-phase domains (APDs). The presence of such APDs is well-
documented in the literature [7, 8, 9], and significant effort has been
directed towards their suppression [2, 10]. As a result, GaP on quasi-exact
(001) Si templates are now commercially available. These templates are free of
dislocations, stacking-faults and possess an APD-free surface with excellent
morphology [11].

In the present contribution, the use of these templates for the growth of GaP
and AlGaP using metalorganic chemical vapor deposition (MOCVD) is
investigated. Regrowth on these templates typically commences with a high-
temperature annealing step in a phosphine ambient to remove the oxide that
forms on top of the template. In the conditions developed by the present group
of authors for the growth of high-quality, defect-free GaP and
$\textnormal{Al}_{0.7}\textnormal{Ga}_{0.3}\textnormal{P}$ layers [12], this
annealing step as well as the subsequent growth of AlGaP are carried out at
$850\text{\,}\mathrm{\SIUnitSymbolCelsius}$. At this temperature, the lattice
mismatch between GaP and Si increases to $0.53\text{\,}\mathrm{\char
37\relax}$. While the mismatch is still low, the critical layer thickness is
bound to be exceeded during growth, leading to the relaxation of strain
through the generation of misfit dislocations.

The objectives of the present paper are, therefore, to assess the mechanical
stability of the GaP/Si template during high-temperature annealing and
subsequent growth, and to evaluate the impact of any defects induced by
plastic relaxation of misfit strain on the growth of AlGaP layers. The paper
is structured as follows: a succinct description of the GaP/Si templates and
the experiments that were carried out is given first. The stability of GaP/Si
templates during annealing and the efficiency of the annealing in removing the
native oxide are then assessed for an array of temperatures. Finally, the
impact that defects induced by the annealing have on the subsequent growth of
GaP is discussed.

##  2 Experiment

A $200\text{\,}\mathrm{mm}$ GaP on Si template was purchased from
$\textnormal{NAsP}_{\textnormal{III/V}}$ GmbH [11]. This template consists of
$50\text{\,}\mathrm{nm}$ of GaP on 001 Si. The Si substrate is almost exact,
with a small intentional miscut of $0.18\text{\,}\mathrm{\SIUnitSymbolDegree}$
in [110]. According to Reference [2], this miscut, in conjunction with
specific growth conditions, allows one to favor one crystal polarity over the
other, resulting in a GaP/Si template with no observable APDs emerging at the
GaP surface. Furthermore, given the small mismatch between GaP and Si - only
$0.36\text{\,}\mathrm{\char 37\relax}$ at room temperature - the GaP layer is
pseudomorphically accommodated on the Si template and, therefore, contains
very few dislocations. AFM images of the surface of GaP as received are shown
in Figure 1. The surface exhibits step flow, with a root mean square (RMS)
roughness of $0.2\text{\,}\mathrm{nm}$ and no visible defects in areas as
large as $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Cross-sections were prepared for observation in a transmission electron
microscope (TEM). The cross-sections for transmission electron microscopy were
prepared from some of the samples using Focused Ion Beam (FIB) ion milling and
thinning. Prior to FIB ion milling, the sample surface was coated with
$50\text{\,}\mathrm{nm}$ of carbon to protect the surface from the platinum
mask deposited used for the ion milling process. Ion milling and thinning were
carried out in a FEI SCIOS dual-beam FIB-SEM. Initial etching was performed at
$30\text{\,}\mathrm{keV}$, and final polishing was performed at
$5\text{\,}\mathrm{keV}$. All TEM cross-sections were observed in an
aberration-corrected FEI TITAN 200 TEM-STEM operating at
$200\text{\,}\mathrm{keV}$. A high-angle annular dark-field image of the
GaP/Si template is shown in Figure 2 (a). The image shows the presence of a
$3\text{\,}\mathrm{nm}$ thick oxide layer on the surface of the
$50\text{\,}\mathrm{nm}$ thick GaP layer. At this scale no defects are present
in the GaP layer. Furthermore, 002 Dark-field TEM images of the samples
confirm that antiphase domains are indeed buried and do not emerge at the
surface of the template - see Figure 2 (b). Nevertheless, misfit dislocations
can be seen at the GaP/Si interface, as shown in the 220 Dark-field TEM image
in Figure 2 (c). The grid has an approximate period of
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The presence of this grid of
dislocations, indicates the onset of plastic relaxation in the GaP/Si
template, despite the small mismatch and relatively thin GaP layer. No
threading dislocations are visible in the lamella, however. Given that the
cross-sections are $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ long by
$80\text{\,}\mathrm{nm}$ wide, the threading-dislocation density is less than
$5\text{\times}{10}^{7}\text{\,}{\mathrm{cm}}^{-2}$.

Figure 1: (a) $100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ and (b)
$4\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ AFM images showing the surface
of GaP on a GaP/Si template procured from
$\textnormal{NAsP}_{\textnormal{III/V}}$ GmbH. Step-flow morphology is
observed, with the step-flow occurring along the 001 direction. The RMS
roughness is $0.2\text{\,}\mathrm{nm}$.

The $200\text{\,}\mathrm{mm}$ template was diced into $50\text{\,}\mathrm{mm}$
wafers using laser cutting. Annealing and growth was carried out in a Veeco
D180 Turbodisk reactor. The annealing experiments were carried out at three
different annealing temperatures were used:
$645\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$745\text{\,}\mathrm{\SIUnitSymbolCelsius}$, and
$845\text{\,}\mathrm{\SIUnitSymbolCelsius}$, for $30\text{\,}\mathrm{min}$ in
a phosphine ambient. The lowest temperature is close to the one disclosed in
Reference [2], discussing the fabrication of the templates. The highest
temperature is similar to one used for the obtained high quality GaP and AlGaP
in Reference [12]. In addition to the annealing experiments, two samples were
grown on the GaP/Si templates using MOCVD: a $200\text{\,}\mathrm{nm}$ GaP
layer to test GaP growth, and, separately, the following structure to test
AlGaP growth: a nominally $325\text{\,}\mathrm{nm}$ thick GaP buffer followed
by two periods consisting of a $47\text{\,}\mathrm{nm}$ thick AlGaP layer,
containing $73\text{\,}\mathrm{at.}\text{\,}\mathrm{\char 37\relax}$ aluminum,
and a $64\text{\,}\mathrm{nm}$ thick GaP layer. Additionally, the
$80\text{\,}\mathrm{nm}$ of the GaP buffer contain five
$1\text{\,}\mathrm{nm}$ thick AlP markers, spaced every
$16\text{\,}\mathrm{nm}$. Both samples were grown at a reactor temperature of
$845\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and a reactor pressure of
$20\text{\,}\mathrm{Torr}$, using trimethylaluminum, trimethylgallium, and
phopshine as precursors to elementary aluminum, gallium, and phosphorous,
respectively.

Figure 2: (a)HAADF-STEM image of the as-received GaP/Si template. The image
shows that the $50\text{\,}\mathrm{nm}$ thick GaP layer is covered by a
$3\text{\,}\mathrm{nm}$ thick oxide layer. (b) 002 Dark-field TEM image
showing the buried antiphase domains. In all the samples tested none of the
domains emerge on the surface. (c) 220 Dark-field TEM image showing
dislocations (black arrows) spaced by $1.5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ along the GaP/Si interface, indicating that despite the low mismatch and
the relatively thin GaP layer, plastic relaxation has set on in the GaP/Si
template

##  3 Results

The four AFM images in Figure 3 show the evolution of the surface morphology
of the GaP/Si template as a function of temperature. At the lowest annealing
temperature, $645\text{\,}\mathrm{\SIUnitSymbolCelsius}$, the GaP layer
preserves it’s surface morphology, exhibiting step-flow and no visible
defects. Nevertheless, the surface still exhibits spheres of oxide, indicating
that the annealing temperature was not sufficient to completely remove the
$3\text{\,}\mathrm{nm}$ oxide present at the surface of the template.
Annealing for an hour at the same temperature did not yield any different
results (not shown here). At higher temperatures, the oxide is completely
removed and the step-flow morphology is still present, but the surface also
exhibits stripes that are between $250\text{\,}\mathrm{nm}$ and
$500\text{\,}\mathrm{nm}$ in width. At the highest annealing temperature,
$845\text{\,}\mathrm{\SIUnitSymbolCelsius}$, these stripes are more markedly
present, creating a $3\text{\,}\mathrm{nm}$ high step. Additionally, several
dislocations emerge at the surface of GaP, proof that the GaP layer
plastically relaxes during the thermal annealing. The estimated threading-
dislocation density is $2\text{\times}{10}^{8}\text{\,}{\mathrm{cm}}^{-2}$, at
least an order of magnitude higher than that present in the template
initially.

Figure 3: $25\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ AFM images of the
surface of the GaP/Si template (a) before, and after annealing at (b)
$645\text{\,}\mathrm{\SIUnitSymbolCelsius}$, (c)
$745\text{\,}\mathrm{\SIUnitSymbolCelsius}$, and (d)
$845\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $30\text{\,}\mathrm{min}$. At
$645\text{\,}\mathrm{\SIUnitSymbolCelsius}$ the atomic steps are preserved,
but the surface is riddled with small pockets of oxide that was not removed.
At higher annealing temperatures, these pockets disappear and the step-flow
morphology is preserved. At $745\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$845\text{\,}\mathrm{\SIUnitSymbolCelsius}$, however, bands that are on
average $375\text{\,}\mathrm{nm}$ wide appear to rise above from the
background. These are particularly visible in the sample annealed at
$845\text{\,}\mathrm{\SIUnitSymbolCelsius}$. At this temperature the elevation
of these bands is $5\text{\,}\mathrm{nm}$. These bands are indicative of
plastic relaxation taking place in the GaP layer of the template during high
temperature annealing.

The defects observed on the annealed samples persist in the
$200\text{\,}\mathrm{nm}$ GaP layer grown on the GaP template, as shown the
AFM images of the sample surface in Figure 4. The threading dislocation
density, estimated from the density of pits in the AFM images, is
$2\text{\times}{10}^{8}\text{\,}{\mathrm{cm}}^{-2}$, no different than that
observed for the sample at $845\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The
GaP surface is also rougher owing to the onset of cross-hatch, with an RMS
roughness of $0.9\text{\,}\mathrm{nm}$, although step-flow is still preserved.

Figure 4: (a) $100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ and (b)
$4\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ AFM images of the GaP surface
after the growth of a $200\text{\,}\mathrm{nm}$ of GaP on the GaP/Si template
using MOCVD. Although step flow is still observed, the surface undulates along
the <010> direction. This undulation is due to crosshatch. Crosshatch in GaP
has been shown to occur anisotropically, more dense along the <010> direction
than the <100> direction [9]. The surface is dotted by small pits where
threading dislocations emerge, pointed out by white arrows in image (b). The
threading dislocation density is estimated at
$1\text{\times}{10}^{8}\text{\,}{\mathrm{cm}}^{-2}$, four orders of magnitude
higher than that of the initial template. In addition to the emergence pits
and the crosshatch, at least one $500\text{\,}\mathrm{nm}$ wide band delimited
by two lines is pointed out by arrows in image (a), indicating that the bands
that appear during annealing persist throughout subsequent growth of GaP.

The defects observed in the GaP sample significantly perturb the growth of
AlGaP, as revealed by the dark-field TEM images of the AlGaP sample, shown in
Figure 5. These images reveal that the stripes observed already during
annealing are bordered by pairs of micro-twins. When these cross the AlGaP
layer, they induce severe faceting, mainly [112] and [114] planes. This
faceting is mainly visible at the AlGaP/GaP interface. Subsequent growth of
GaP tends to smooth this roughness, as seen at the second GaP/AlGaP interface
in Figure 5, but the surface becomes again rougher during the growth of the
second AlGaP layer. The RMS roughness at the surface is
$5\text{\,}\mathrm{nm}$, more than an order of magnitude higher than the
initial roughness of the template.

Figure 5: (a) 220 dark-field TEM image of the AlGaP/GaP structure grown on the
GaP template. A dense grid of misfit dislocation is present at the GaP/Si
interface, indicating that the structure is fully plastically relaxed. In
addition to the dislocation grid, micro-twins appear in the structure. These
often appear in pairs spaced by $365\text{\,}\mathrm{nm}$, as revealed in (b)
showing a 220 dark-field TEM image close up of one such pair. These pairs are
linked to . These micro-twins induce faceting in AlGaP, significantly
roughening the GaP/AlGaP interfaces. This faceting is highly detrimental to
the optical properties of AlGaP/GaP pairs. As shown in Figure 3, these micro-
twins already appear during the annealing of the GaP/Si template.

Finally, the GaP/Si interface now exhibits a dense misfit-dislocation grid,
with an approximate period of $30\text{\,}\mathrm{nm}$. Upon close inspection,
this dislocation grid is composed up to $60\text{\,}\mathrm{\char 37\relax}$
of edge dislocations, and $40\text{\,}\mathrm{\char 37\relax}$ partial
dislocations. Half of the partial dislocations at the interface combine with a
Lomer-Cottrel dislocation in Si to form a stair-rod dislocation [13]. This
type of defect is more common in highly mismatched systems, such as Ge/Si that
has a lattice mismatch of $4\text{\,}\mathrm{\char 37\relax}$, an order of
magnitude higher than GaP/Si [14], and to the extent of our knowledge has not
been reported for lowly mismatched systems. The stair-rod itself appears as an
inverted triangle at the GaP/Si interface. One such defect is shown in Figure
6 (a). Two apexes of the the triangle are partial misfit dislocations at the
GaP/Si interface. Tracing the Burgers’ circuits of the partial dislocations -
Figures 6 (b) and (c) - reveals that their Burgers’ vector is
$\mathbf{b}=\frac{1}{6}\mathbf{a}_{112}$, in keeping with theory. Stacking
faults extend from them and slip along [111] planes in Si until they meet at
about $30\text{\,}\mathrm{nm}$ from the GaP/Si interface. There they combine
to form the Lomer-Cottrel dislocation, that has a Burgers’ vector of
$\mathbf{b}=\frac{1}{2}\mathbf{a}_{110}$ and is sessile.

Figure 6: (a) HAADF-STEM image of a the GaP/Si interface in the AlGaP sample
showing a stair-rod dislocation. The stair-rod forms at the junction of the
stacking faults (SF1 and SF2) that extend from two partial dislocations (PD1
and PD2) at the GaP/Si interface and meet in the Si substrate, forming a
sessile Lomer-Cottrel dislocation. (b,c) Insets showing magnified portions of
the HAADF-STEM image in (a), centered around PD1 and PD2, and their Burgers’
circuits.

##  4 Discussion

Both AFM and TEM results show that the $50\text{\,}\mathrm{nm}$ GaP layer in
the template does not contain any emerging APDs, and presents a smooth surface
with a step-flow morphology and a roughness of only $0.2\text{\,}\mathrm{nm}$.
Nevertheless, the TEM images reveal that even at this thickness, which is
lower than the critical thickness for plastic relaxation of GaP, reported at
$90\text{\,}\mathrm{nm}$ [15], a loose misfit-dislocation grid is already
present at the GaP/Si interface. In the literature, this has been associated
with thermal-stress induced relaxation of GaP during the cooling down phase of
the GaP/Si template following GaP growth at temperatures higher than
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The reported threading
dislocation densities in optimized conditions were of the order of
$1\text{\times}{10}^{7}\text{\,}{\mathrm{cm}}^{-2}$ [16, 17], in agreement
with the estimate obtained from TEM on the reference sample.

In two samples grown by MOCVD, the total GaP thickness is more than
$250\text{\,}\mathrm{nm}$, exceeding the critical thickness. In the literature
full plastic relaxation of misfit strain that is due to lattice mismatch
between GaP and Si is not achieved until a thickness of
$200\text{\,}\mathrm{nm}$ [17, 15, 18]. At the reported thickness thickness
the samples are, therefore, fully plastically relaxed, and, accordingly,
exhibit dislocation densities in the
$1\text{\times}{10}^{8}\text{\,}{\mathrm{cm}}^{-2}$ range. The plastic
relaxation of the strain induced to mismatch is further supported by the
appearance of Lomer-Cottrel locks in Si. Indeed, the appearance of misfit
dislocations in the Si substrate has been previously associated with the
relaxation of strain that is due to the lattice mismatch, whereas thermal
stress tends to generate defects at the GaP/Si interface or in GaP [16].
Although the RMS roughness of GaP sample increases, it still remains within
reasonable limits, i.e. less than $1\text{\,}\mathrm{nm}$. This roughness
could be improved by increasing the V/III ratio and/or the reactor pressure.

The most cumbersome defects, however, are the micro-twins that already appear
during high-temperature annealing of the GaP/Si template. These defects induce
heavy faceting in the AlGaP layer, rendering these layers inappropriate for
the growth of AlGaP/GaP Bragg mirrors. The micro-twins reported here are
denser when the annealing temperature is increased. This result is contrary to
what has been observed in the case of GaP grown on
$4\text{\,}\mathrm{\SIUnitSymbolDegree}$ off-cut Si using migration-enhanced
molecular-beam epitaxy [19]. Indeed, in that reference, micro-twins are
eliminated by implementing an annealing step after the initial nucleation of a
$20\text{\,}\mathrm{nm}$ GaP layer. It is unclear whether the miscut is solely
responsible for this difference, or whether the micro-twins could be
annihilated by further tweaking of the growth and annealing conditions on the
current templates. Alternatively, growth could be performed in a temperature
range closer to what has been reported by NAsP${}_{\textnormal{III/V}}$ Gmbh.,
i.e. between $450\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$600\text{\,}\mathrm{\SIUnitSymbolCelsius}$ [2], using precursors that are
better suited to low-temperature growth, such as triethylgallium and tri-tert-
butylphosphine. Nevertheless, incorporation of high amounts of aluminum at low
temperatures may prove challenging.

##  5 Conclusion

Despite excellent surface properties and the absence of emerging APDs, the
GaP/Si templates used in the present contribution are revealed not to be
compatible with the growth conditions developed of high-quality AlGaP and GaP
growth on GaP substrates. Indeed, the templates are unstable at high
temperatures, plastically relaxing to form micro-twins that detrimentally
affects the growth of AlGaP in particular. Furthermore, the oxide layer
present at the surface of the templates is not readily removed through
annealing in a phosphine ambient at lower temperatures. A combination of wet
etching to remove the oxide and growth with alternative metalorganic
precursors that are better suited to low-temperature growth may be the only
solution towards growth of high quality GaP and AlGaP on these templates.

## Acknowledgements

The authors would like acknowledge J. Nagle, A. De Rossi, and S. Combrié for
their support and fruitful discussions. The authors thankfully acknowledge
funding from the CNRS Renatech network and the ANR Labex TEMPOS
(ANR-10-EQPX-0050).

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