11institutetext: University of British Columbia, Kelowna BC V1V1V7, Canada
11email<EMAIL_ADDRESS>

# A Novel Bounding Box Regression Method for Single Object Tracking

Omar Abdelaziz 11 0009-0009-2416-5700 Mohmaed Sami Shehata 11
0000-0003-1289-0285

###### Abstract

Locating an object in a sequence of frames, given its appearance in the first
frame of the sequence, is a hard problem that involves many stages. Usually,
state-of-the-art methods focus on bringing novel ideas in the visual encoding
or relational modelling phases. However, in this work, we show that bounding
box regression from learned joint search and template features is of high
importance as well. While previous methods relied heavily on well-learned
features representing interactions between search and template, we hypothesize
that the receptive field of the input convolutional bounding box network plays
an important role in accurately determining the object location. To this end,
we introduce two novel bounding box regression networks: inception and
deformable. Experiments and ablation studies show that our inception module
installed on the recent ODTrack outperforms the latter on three benchmarks:
the GOT-10k, the UAV123 and the OTB2015.

###### Keywords:

Single object tracking, Bounding box regression, One-stream framework, Vision
transformers.

##  1 Introduction

Single object tracking is a challenging computer vision subfield that is
ubiquitous in many applications such as video surveillance and augmented
reality. The challenging nature of tracking is associated with the problem
definition as well as the techniques to address the problem. Given a sequence
of images, it is required to identify the size and location of an object given
by a bounding box annotation in the first frame of the sequence. In single-
object tracking, the object is also identified by a bounding box.

The current prevalent pipeline in single object tracking is the joint feature
extraction and relation modelling pipeline introduced in [7] and [35].
Exploiting Vision Transformers (ViTs), this pipeline leverages the natural
attention capabilities in relationship association to locate the template
object in the search frame in a low-dimensional space. A bounding box
prediction head is used afterwards to map the transformer attention map into
bounding box coordinates.

Despite showing remarkable success, the current tracking methods are focused
on the feature extraction and relation modelling phases in a one-stream manner
while being less attentive to the bounding box regression phase. Bounding box
regression is usually done using a simple multi-branch convolutional network.
These networks simplify the bounding box regression by dividing it into
classification and regression subproblems. The classification stage aims to
separate the object from the background. In contrast, the regression branch
aims to predict four float values that represent the coordinates of the top-
left corner and either the width and the height or the coordinates of the
bottom-left corner of the box. This simple design depends on the number of
parameters of the Vision Transformer [10] to learn a good embedded
representation that overcomes problems in the bounding box network.

We hypothesize that the simple convolutional pipeline of bounding box networks
is not enough to capture complex global and local relations in the embedded
feature representation output by the transformer. We introduce two simple yet
effective bounding box networks that are capable of learning a variable
receptive field to enable a more accurate determination of the target
location. In particular, we introduce the Inception and the deformable
convolution Inception bounding box networks based on the Inception [25] and
deformable convolutional concepts [37]. We show backed by experiments that our
bounding box networks can capture more useful information from the ViT
embeddings.

To this end, the contributions of this work are three-fold:

  * •

We propose a new bounding box head for ViT-based single object trackers,
namely, the Inception module that addresses the fixed local receptive field of
the bounding box network of single object tracking transformers.

  * •

We propose another variant of the Inception module, the deformable convolution
Inception module, which learns the required receptive field for a better-
performing bounding box network.

  * •

We conduct extensive experiments and ablation studies to show the
effectiveness of our proposed modules. Our best model sets a new state-of-the-
art.

##  2 Related Works

###  2.1 Tracking Paradigms

Recent deep-learning-based single-object trackers involve either CNN or ViT
backbones. CNN paradigm has been dominated by the Siamese architecture since
their introduction in [1, 26]. The basic idea of cross-correlation in the
embedding space employed in [1] is employed in many variants [6, 18, 17, 29].
Siamese trackers aim to embed the template object and the search image in a
well-chosen space where the cross-correlation between the embedded
representations results in a score map representing the object’s location.
Finally, post-processing techniques such as the cosine window penalty have to
be adopted to calculate the bounding box coordinates from the correlation
score map. Despite showing remarkable success, this paradigm had several
drawbacks. First, the simple cross-correlation operation assumes high-quality
feature embeddings, and this is not guaranteed by simple CNN as AlexNet [16]
or VGGNet [21] that were used as the feature extractor. Second, the complex
post-processing steps required in the bounding box coordinates calculation
phase may hinder the performance if its parameters are not chosen optimally.
In our work, we hypothesize that bounding box coordinates must be learned as a
part of an end-to-end learning framework without a hand-crafted set of
thresholds or parameters to achieve the robustness and accuracy of bounding
box calculation. Our method not only learns the coordinates of the bounding
box directly but also the receptive field of the bounding box CNN, thus,
resulting in extracting more useful information from the previous phase.

The CNN paradigm was dominant until the ViT-based trackers were introduced [5,
28]. These trackers reimplemented the dominant Siamese tracking with a
transformer visual encoder. Other research leveraged the powerful attention
capabilities to bring closer end-to-end ViT tracking. Several works
simultaneously introduced joint feature extraction and relation modelling [3,
35, 7]. Generally, this framework assumes that the transformer is capable of
extracting the visual features alongside learning the relationships between
the embedded representations. After obtaining a representative feature map
from the ViT, these methods apply a convolutional bounding box head to predict
the bounding box coordinates mainly in the classification and regression
phases. This one-stream paradigm has been dominant with many works that
propose changes to ViT architecture itself to adapt to the single object
tracking problem [11, 22, 4, 30].

###  2.2 Convolutional Design Choices

####  2.2.1 Inception network

The inception network was introduced in [21] to overcome the problem of the
narrow receptive field of previous CNNs. The main idea behind the original
paper and its variants [23] is that for a convolutional network to go deeper,
its receptive field has to be wide enough to process the input on many deep
stages without losing information from the early stages. For that purpose, the
Inception module carries on several convolutional passes in parallel, each
representing a different convolutional filter. Different outputs of the passes
are concatenated together and passed to a $1\times 1$ convolutional filter to
reduce the channel dimension in a learnable manner. In this work, we apply the
simplest inception module to the output of a ViT encoder from three known
trackers.

####  2.2.2 Deformable Convolutions

Deformable convolution [8] is an extension of the regular convolution where
the sampling positions of the convolutional filter are learnable. In this
regard, the deformable convolution operation involves two runs of
convolutional filters. The first run determines the offsets that the filter of
the second run will use to sample values of the input feature map, which will
apply the convolution operation. In another version of the deformable
convolutions [37], the amplitudes of the input feature maps are modulated by a
learnable mask, ensuring an extra degree of freedom. We implement a variant of
the proposed Inception bounding box network with deformable convolutions to
exploit the powerful learning paradigm encapsulated by the learnable sampling
positions and input amplitudes.

##  3 Methodology

SearchTemplate Transformer
Model Bounding Box Model Downsampling CNN Redesigned Inception Block
Downsampling CNN Redesigned Inception Block Downsampling CNN Redesigned
Inception Block Center
Map Size
Map Offset
Map

Figure 1: The overall architecture of the recent ViT-based single object
trackers.

Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}1\times
1}$Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}1\times
1}$Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}1\times
1}$AVG
PoolConv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}1\times
1}$Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}3\times
3}$Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}3\times
3}$Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}3\times
3}$Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}1\times
1}$

Figure 2: The simple Inception subnetwork architecture adapted from [24]. The
blue-filled circle indicates concatenation operation. Filter sizes are shown
in the figure. Please refer to [24] for more details about the parameters of
each convolution.

Conv$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}1\times
1}$ Deform
$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}3\times
3}$ Regular
$\displaystyle{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}3\times
3}$

Figure 3: The deformable inception block.

###  3.1 Adopted Tracking Framework

We hypothesize that modern ViT-based single-object trackers, such as OSTrack
[35] and ODTrack [36] lack good exploitation of the joint search and template
embeddings to extract the location information.

Given a ViT-based backbone, the output after the joint feature extraction and
relation modelling phase is an embedding tensor of dimensions $(B,H\times
W,D)$ where $B$ is the batch size, $H\times W$ is the dimension of the
transformer patch, and $D$ is the embedding dimension. For the ViT-based
trackers represented by the three selected trackers, this output tensor is
subjected to a CNN-based bounding box regression network that extracts three
score maps representing the center location, the size, and the offset of the
center of the bounding box as shown in Fig. 1. Given that the input tensor
might contain global correlation as well as local correlation, we hypothesize
it is essential that the bounding box network has multiple receptive fields
that enable the network to capture relationships on both the global and local
levels.

###  3.2 Inception Network

To this end, we propose the usage of an Inception block [25] in two
alternatives as depicted in Fig. 2 and Fig. 3.

The simpler inception block is adopted from [24]. As depicted in Fig. 2, it is
composed of four parallel convolutional paths. The four convolutional paths
share the functionality of mapping the input feature map to a new feature
space while each of them has a different receptive field. Namely, while a
bigger filter size of $7\times 7$ that is realized by the two $3\times 3$
convolutions shown in the upper-most branch foster a more global understanding
of attentional relationships in the input tensor, the smaller ones ($1\times
1$, $3\times 3$) focus on a local context. After feeding the embedding tensor
into the convolutional layers, the obtained outputs are concatenated and fed
into a $1\times 1$ convolutional filter to reduce the number of channels to
the original input number of channels, hence, speeding up the computations.

The more sophisticated inception block is based on the same idea but with a
different implementation. Given the intrinsic adaptive receptive field that
Deformable Convolutional Networks have [37], we design a deformable-based
inception block of convolutions that have filter size of $3\times 3$ where
each filter performs deformable convolution operation and the final output is
concatenated and fed to a regular $1\times 1$ convolution as shown in Fig. 3.
The deformable convolutions enable a larger receptive field for the same
filter size by learning the sampling positions of the input. In addition to
the deformable convolution, we add a branch of regular convolution to consider
local contiguous context, therefore simplifying the learning objective and
decreasing convergence time.

##  4 Experiments

Table 1: Performance comparison against the state-of-the-art methods. We
emphasize that our models are only trained on the training split of the
GOT-10k dataset. Significant results are in bold

Method | GOT-10k | UAV123 | OTB2015
---|---|---|---
AO | $SR_{0.5}$ | $SR_{0.75}$ | AUC | AUC
SiamFC [1] | $34.8$ | $35.3$ | $9.8$ | $46.5$ | $58.2$
ATOM [9] | $55.6$ | $63.4$ | $40.2$ | $65.0$ | -
DiMP [2] | $61.1$ | $71.7$ | $49.2$ | $65.4$ | $68.4$
SiamRCNN [27] | $64.9$ | $72.8$ | $59.7$ | $64.9$ | $70.1$
Stark [34] | $68.8$ | $78.1$ | $64.1$ |  | $69.1$
---
$68.5$
SBT-B [32] | $69.9$ | $80.4$ | $63.6$ | - | $70.9$
Mixformer [7] | $70.7$ | $80.0$ | $67.8$ | $70.4$ | -
OSTrack [35] | $73.7$ | $83.2$ | $70.8$ | $\mathbf{70.7}$ | -
AiATrack [12] | $69.6$ | $80.0$ | $63.2$ | $70.6$ | $69.6$
SeqTrack [4] | $74.5$ | $84.3$ | $71.4$ | $68.6$ | -
GRM [13] | $73.4$ | $82.9$ | $70.4$ | $70.2$ | -
VideoTrack [33] | $72.9$ | $81.9$ | $69.8$ | $69.7$ | -
ARTrack [30] | $75.5$ | $84.3$ | $74.3$ | $70.5$ | -
ODTrack-B [36] | $77.0$ | $\mathbf{87.9}$ | $75.1$ | - | $\mathbf{72.3}$
ODTrack-B (Our machine) | $75.6$ | $86.1$ | $72.7$ | $65.8$ | $68.5$
ODTrack-B-Inception (Ours) | $\mathbf{77.3}$ | $87.5$ | $\mathbf{75.5}$ | $65.8$ | $69.0$

Table 2: Investigation of the performance of the two proposed bounding box
networks on three modern trackers. Significant results are in bold. All three
trackers are retrained from MAE [14] pretraining weights on our machine using
the same configurations provided in their repositories for fair comparison.

Method | GOT-10k | UAV123 | OTB2015
---|---|---|---
AO | $SR_{0.5}$ | $SR_{0.75}$ | AUC | AUC
OSTrack-B | $70.1$ | $79.0$ | $66.3$ | $65.79$ | $66.68$
OSTrack-B-Inception | $71.6$ | $80.7$ | $66.9$ | $66.01$ | $68.18$
OSTrack-B-deformable | $70.8$ | $79.8$ | $67.1$ | $\mathbf{67.78}$ | $66.98$
ODTrack-B | $75.6$ | $86.1$ | $72.7$ | $65.79$ | $68.46$
ODTrack-B-Inception | $\mathbf{77.3}$ | $\mathbf{87.5}$ | $\mathbf{75.5}$ | $65.79$ | $\mathbf{69.03}$
ODTrack-B-deformable | $76.8$ | $87.2$ | $73.3$ | $64.30$ | $68.89$

Table 3: Study of the performance of ODTrack with deformable convolution
Inception module. "deformable only" refers to the deformable branch without
the regular convolution while "Conv only" refers to the original ODTrack [36].
"Both" refers to using the bypass branch. Significant results are in bold.

Method | GOT-10k
---|---
AO($\%$) | $SR_{0.5}(\%)$ | $SR_{0.75}(\%)$
deformable only | $75.4$ | $86.6$ | $\mathbf{73.3}$
Conv only | $75.6$ | $86.1$ | $72.7$
Both | $\mathbf{76.8}$ | $\mathbf{87.2}$ | $\mathbf{73.3}$

###  4.1 Implementation Details

We adopt the ViT-base [10] as the visual encoder with MAE pretraining weights
[14] to train OSTrack [35] and ODTrack [36] using our proposed bounding box
regression networks. The template image shape is $192\times 192\times 3$ while
the search image shape is $384\times 384\times 3$. All models are trained for
$100$ epochs solely on the GOT-10k dataset [15]. Training is conducted on a
server with two $50$ GB GPUs. The batch size is set to $8$. AdamW [19] is used
to optimize the training objective. We sample $40000$ examples per epoch to
train networks based on ODTrack following the configurations found in their
code repository, while the number of samples per epoch for OSTrack-based
networks is $60000$. We keep the learning rate and the weight decay that each
model was originally trained with. We report the performance of the two
proposed variants, namely the Inception module and the Deformable Inception
module. The basic convolutional block used in both of them is a stack of conv,
ReLU, and BatchNorm. The filter details of the convolutions for each variant
are shown in Fig. 2 and Fig. 3.

###  4.2 State of the Art Comparison

We compare our method to $14$ methods on three leading datasets, GOT-10k [15],
UAV123 [20] and OTB2015 [31]. The results shown in Table 1 indicate that our
method variants outperform the state-of-the-art techniques on the three
datasets when compared with retrained models on the same machines.

####  4.2.1 GOT-10k

Generic Object Tracking dataset is a large-scale single object tracking
dataset. It contains $9334$ training sequences, $180$ validation sequences and
$180$ test sequences. The testing protocol of the dataset requires models to
be trained only on the GOT-10k training sequences. The training and test
sequences are not of overlapping classes. As reported in Table 1, our best
performant variant outperforms the state-of-the-art ODTrack-base [36] trained
on our machine by $1.7\%$ in Average Overlap (AO). Moreover, for the Success
Rate at $0.5$ IoU threshold ($SR_{0.5}$) and the ($SR_{0.75}$), our model
outperforms ODTrack by $1.4\%$ and $2.8\%$ respectively.

####  4.2.2 UAV123

UAV123 is a aerial viewpoint video dataset that contains $123$ challenging
sequences. It contains training and testing sequences. However, we only use
the testing set for evaluation purposes without training on the training set.
According to the shown results in Table 1, our method outperforms the aligned
state-of-the-art techniques while it is only trained on the GOT-10k training
split.

####  4.2.3 OTB2015

Object Tracking Benchmark [31] is a pioneering testing set for single object
tracking that consists of $100$ challenging frames collected from YouTube. The
dataset labels are publicly available. The performance pattern on this dataset
is similar to the one in the GOT-10k as they are both datasets of the same
natural image domain.

###  4.3 Ablation Study

We examine the effect of the proposed simple Inception and deformable
convolution Inception on the two adopted modern trackers stated in section 3.
As reported in Table 2, our Inception and deformable inception methods
outperform the locally trained models on the GOT-10k test set considering the
Average overlap by $1.5\%$ and $0.6\%$ respectively. The pattern is consistent
despite the different tracking ideas implemented in the different modules,
indicating the effectiveness of the inception and deformable inception
convolutional networks in learning a suitable receptive field to process the
joint search and template embeddings. Notably, the pure Inception is more
powerful enhancing the performance on all three datasets. This indicates the
higher learning capacity of the inception module compared to the deformable
inception. We also investigate the effect of the skip branch that has the
$3\times 3$ convolutional filter of the deformable convolution Inception
module in Table 3. It can be shown that the adopted variant is the best
performant deformable module indicating the importance of the $3\times 3$
convolution skip connection in facilitating the learning objective and
increasing model capacity.

###  4.4 Visualization

Figure 4: Success cases of the proposed methods on the OTB2015 Basketball
sequence. It can be noted that the pure Inception network is most accurate
especially when comparing the first column of frames. Although bounding boxes
converge at frame $\\#50$, the initially predicted bounding boxes at frame
$\\#2$ indicate the fast adaptation of the Inception bounding box network.

We qualitatively visualize the output of our two bounding box regression
networks in Fig. 4. As depicted, the regular Inception has a better bounding
box quality. This indicates the high model capacity of the inception module
that enables the best exploitation of the visual features given by the ViT in
ODTrack [36].

##  5 Conclusion

In this work, we show the significance of the bounding box regression given
the visual interaction features of a one-stream tracking transformer. We show
that a bounding box-regression network lacking a suitable receptive field can
hinder the performance of a transformer tracking model due to the lack of good
exploitation of the visual features provided by the backbone. Experiments show
that our methods of Inception and deformable inception bounding box networks
perform favourably against state-of-the-art ODTrack by $1.7\%$ of AO on the
GOT-10k [15] dataset. As future pathways, this work can be extended to ablate
the different emerging CNN techniques and find the CNN bounding box network
that strikes the balance between a good inductive bias and a global receptive
field.

####  5.0.1

The authors have no competing interests to declare that are relevant to the
content of this article.

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# Finding the stable structures of 2D hexagonal materials with Bayesian
optimization: Beyond the structural relationship with 3D crystals in weakly-
bonded binary systems

Shota Ono<EMAIL_ADDRESS>Department of Electrical, Electronic and
Computer Engineering, Gifu University, Gifu 501-1193, Japan

###### Abstract

The graphene-graphite relationship in structural geometry is a basic principle
to predict novel two-dimensional (2D) materials. Here, we demonstrate that
this is not the case in binary metallic systems. We use the Bayesian
optimization framework combined with the density-functional theory approach to
determine the stable configuration of atomic species on a hexagonal plane. We
show that the optimized structure of 2D Cu-Au exhibits the hexagonal lattice
of a hexagonal ring of Cu atoms containing one Au atom, where the number of
the Cu atoms is larger than that of the Au atoms in the unit cell, which is
difficult to speculate from the atomic distribution of CuAu in the L10
structure. We also show that 2D Cu-$X$ with $X=$ Be, Zn, and Pd have hexagonal
or elongated rings containing different atoms in the unit cell. Based on the
binary Lennard-Jones model, we propose that such structures can appear for
weakly-bonded systems located in between the phase-separated and strongly-
bonded systems with the interatomic interaction energy between different
species.

##  I Introduction

A large family of two-dimensional (2D) materials has been discovered
experimentally and proposed theoretically. A basic principle to find novel 2D
materials is the structural relationship between 2D and 3D systems. This
originates from the successful synthesis of graphene exfoliated from its bulk
counterpart, graphite graphene . This concept has been used to predict 2D
materials by calculating the exfoliation energy of 2D layers from the 3D
counterparts ashton , predicting more than 600 potential 2D materials, whereas
many of them are left to be synthesized. Nevalaita and Koskinen have applied
similar analogy to elemental metals nevalaita : by considering the fact that
the 3D elemental metals have the close-packed structures (the fcc or hcp) as
the ground state, the 2D elemental metals are expected to form the hexagonal
structure. They studied the stability of 2D metals in the planar hexagonal
(PHX), square, and honeycomb structures by performing the density-functional
theory (DFT) calculations. They have demonstrated that among three structures
the PHX structure is the lowest energy state, and that the larger the cohesive
energy in the 3D phase, the larger that in the 2D phases, which establishes
the 2D-3D structural relationship in elemental metals. More recently, the
dynamical stability and magnetic property of elemental metals in the PHX
structure have been studied using first-principles approach ono2020 ; ren .

It has not been understood whether the stable structure of 2D systems is
analogous to that of 3D systems for binary systems $X$-$Y$, where $X$ and $Y$
are the metallic elements in the periodic table. This is because in the 3D
bulk the $X$-$Y$ systems can form a wide variety of ordered structures
depending on the composition ratio of atoms. While the author systematically
studied the energetic stability of the binary 2D metals $XY$ (46 metallic
elements of $X$ and $Y$) ono_satomi  and the dynamical stability of Cu$X$
ono_meta  and noble metal-based ordered alloys ono2021PRM , only simple
structures such as the buckled honeycomb structure were assumed as potential
phases.

As a model system of binary intermetallic compounds, the 3D Cu-Au has been
extensively studied for many years. While the Cu and Au have the fcc structure
as its ground state, the Cu-Au system forms several ordered structures
depending on the mixing ratio: CuAu in the L10 structure and Cu3Au and CuAu3
in the L12 structure. The DFT results were consistent with such observations,
while the underestimation of the formation energy by a factor of two in the
Cu-Au system and the missing of the L12 CuAu3 along the tie line in the phase
diagram have been reported ozolins ; nonlocalpbe ; isaacs ; ruzsinszky2019 ;
ruzsinszky2020 .

Recently, other ordered phases have been reported for Cu-Au systems. By using
the DFT approach, Pandey et al. showed that the Cu-rich and Au-rich phases
prefer the L12\- and L10-derived structures, respectively pandey . They
attributed these trends to the atomic radius difference: the Au atom (the
atomic radius of 1.44 Å) is bigger than the Cu atom (1.28 Å). Based on this
fact, they interpreted that the Cu-rich structures have a large strain when
the Au and Cu monolayers are stacked in a similar manner of the L10 structure.
In contrast, the Au-rich structures can accommodate the smaller Cu layers, and
form the L10 structure. More recently, the surface morphology of Cu-Au solid
solutions and Cu3Au(100) have been investigated by Liu et al. liu  and Li et
al. li , respectively.

By performing phonon dispersion calculations, the author showed that the CuAu
in the Bh (WC-type) and L11 (CuPt-type) structures are dynamically stable
ono_meta . The Bh and L11 structures are constructed from, respectively, the
ABAB and ABC stacking of the Cu and Au hexagonal layers that are also
dynamically stable ono2020 , suggesting that the 2D layers serve as building
blocks for the 3D structures. Zagler et al. reported the experimental
synthesis of buckled honeycomb CuAu, where the hexagonal Au monolayer is
stacked on the hexagonal Cu monolayer zagler . This can be regarded as a 2D
counterpart for the 3D metastable structures above.

With a wide variety of 3D structures observed in the Cu-Au systems, the focus
of this paper is on the 2D structures. In the present work, we explore the
stable structures of 2D Cu-Au by assuming the hexagonal structure including 16
atoms in the unit cell, i.e., $4\times 4$ supercell. These systems have a vast
amount of configurations for the atomic species ($2^{16}$), so that the
structure optimization of all configurations is a difficult issue. We use the
Bayesian optimization (BO) approach to search for the lowest formation energy
configuration, and find that the optimized structure exhibits the hexagonal
lattice of a hexagonal ring of Cu atoms containing one Au atom, which may be
difficult to speculate from the L10 structure that is the lowest formation
energy structure of 3D CuAu. We also use the BO approach to search for the
stable structure in the binary systems of Cu-$X$, and find that the cases of
$X=$ Be, Zn, and Pd also have hexagonal or elongated rings containing
different atoms in the unit cell. With the binary Lennard-Jones (BLJ) model
calculations, we propose that such structures can appear for weakly-bonded
systems located in between the phase-separated and strongly-bonded systems
with increasing the interatomic interaction energy between different species.
The present results provide interesting examples beyond the graphene-graphite
relationship in structural geometry.

In computational materials design, the BO approach has been applied to
optimize various quantities such as thermal conductivity in solids seko  and
nanostructures shiomi , chemical composition of magnetic compounds fukazawa ,
and foreign atom adsorption in graphene dieb  and GaN(0001) surfaces kusaba .
The present work provides another example of the BO applications.

##  II Computational details

###  II.1 First-principles calculations

Our calculations are based on the DFT within the generalized gradient
approximation pbe . The ultrasoft pseudopotentials was generated by using the
pslibrary dalcorso . The cutoff energy for the wavefunction and charge density
was set to be 60 Ry and 600 Ry, respectively. The Monkhorst-Pack (MP) $k$ grid
of $6\times 6\times 1$ MP  and the smearing parameter of 0.02 Ry smearing
were used to construct the phase diagram (Fig. 3) of the Cu-Au system. The
interlayer distance was fixed to 15 Å to avoid spurious interactions in the
periodic boundary condition. Spin-unpolarized calculations were performed
throughout the paper. Calculations were performed using the Quantum ESPRESSO
qe .

To study the stability of 2D $X$-$Y$, we calculated the formation energy per
atom defined as

$\displaystyle\Delta
E_{j}(X_{n}Y_{m})=\frac{E_{j}(X_{n}Y_{m})-\left[nE_{0}(X)+mE_{0}(Y)\right]}{n+m},$
(1)

where $E_{0}(X)$ and $E_{0}(Y)$ are the total energy of elements $X$ and $Y$
in the PHX structure, respectively. $E_{j}(X_{n}Y_{m})$ is the total energy of
$X_{n}Y_{m}$ in the configuration $j$ with $n+m=16$. Several definitions for
the formation energy have been used to study the 2D materials: the total
energy of the mixed phase is subtracted by the sum of the total energy of
atoms sahin , 3D structures ono_satomi , or 2D structures weng ; ono2021PRM .
The Eq. (1) is the same as that used in Ref. ono2021PRM , where it can give a
negative value of the formation energy for the buckled honeycomb-structured
CuAu that has been synthesized experimentally zagler .

When $n+m=16$, the total atomic configuration is $2^{16}=65536$, and the
computational cost for investigating all configurations might be large at the
DFT level. We thus remove the equivalent configurations by considering the
translational, rotational, and inversion symmetry, and found 7279
configurations to be investigated. However, it is still a large number of
configurations to perform DFT calculations.

To tackle with the optimization problem above, the BO was adapted to search
for the lowest formation energy structure of $X$-$Y$. We used the scikit-
optimize 0.9.0 skopt  to perform the BO for the configurations of atomic
species for the binary 2D systems $X$-$Y$. In the present work, each
configuration was labeled by a sequence of 16 binary numbers such as
0000111100001111, where 0 and 1 indicates the atom $X$ and $Y$, respectively,
and the $l$th figure from the left specifies the atomic species at the $l$th
site in the unit cell on the PHX structure. The BO was done by constructing
the Bayesian statistical model for the formation energy by using Gaussian
process regression, where the Matern kernel was used for the Gaussian process
estimator. A combination of three acquisition functions, including the
expected improvement, the lower confidence bound, and the probability of
improvement, were used, specifying which sample 16 binary numbers next. More
than 100 steps for three runs with different random seeds were performed by
assuming that the objective function is noise-free. The lattice constant $a$
and the atomic position were optimized during each step. The initial guess of
the lattice constant was set to be $a(X_{n}Y_{m})=[na(X)+ma(Y)]/(n+m)$, where
$a(X_{n}Y_{m})$, $a(X)$, and $a(Y)$ are the lattice constant of the
$X_{n}Y_{m}$, $X$, and $Y$ in the PHX structure. The magnitude of $a(X)$ and
$a(Y)$ is four times larger than that of the primitive cell for the elemental
metals in the hexagonal structure (see Table I in Ref. ono2020 ).

To reduce the computational costs, the $4\times 4\times 1$ $k$ grid was used
in the configuration search based on the BO. We have confirmed that the values
of $\Delta E_{j}($CunAu${}_{m})$ are the same as those calculated by the
$6\times 6\times 1$ $k$ grid within an error of a few meV per atom, and the
total energy alignment between different five models (in Fig. 1 below) is not
changed.

Figure 1: Models of $X_{8}Y_{8}$ in the PHX structure (chain, zigzag, zigzag2, stripe, and diamond) and the optimized structure of Cu9Au7 obtained by the Bayesian optimization, where the blue and yellow balls indicate the Cu and Au atoms, respectively. For the relaxed geometry of the optimized structure (right), the atoms are displaced toward the gray balls. The unit cell is indicated by colored red.  Table 1: The $\Delta E_{j}$ (meV/atom) of the $X_{n}Y_{m}$ systems for the five structures with $n=m$ (see Fig. 1) and the four structures with $n\neq m$ derived from the BO calculations. The lowest formation energy in an $X$-$Y$ is shown in bold. $X$-$Y$ | chain | zigzag | zigzag2 | stripe | diamond | $X_{9}Y_{7}$ | $X_{7}Y_{9}$ | $X_{10}Y_{6}$ | $X_{6}Y_{10}$
---|---|---|---|---|---|---|---|---|---
Cu-Ag | $72$ | $62$ | $68$ | $58$ | $58$ | $60$ | $65$ | $61$ | $61$
Cu-Be | $-18$ | $-25$ | $-22$ | $25$ | $5$ | $11$ | $-29$ | $-14$ | $-13$
Ag-Au | $-37$ | $-27$ | $-32$ | $-15$ | $-21$ | $-28$ | $-25$ | $-29$ | $-27$
Cu-Pd | $-46$ | $-43$ | $-44$ | $-19$ | $-28$ | $-45$ | $-39$ | $-51$ | $-32$
Cu-Pt | $-65$ | $-32$ | $-49$ | $0$ | $-6$ | $-7$ | $-52$ | $-23$ | $-44$
Cu-Zn | $-69$ | $-55$ | $-62$ | $-21$ | $-41$ | $-63$ | $-45$ | $-76$ | $-48$
Cu-Au | $-66$ | $-70$ | $-68$ | $-24$ | $-46$ | $-78$ | $-42$ | $-73$ | $-54$
Cu-Ga | $-255$ | $-253$ | $-249$ | $-126$ | $-182$ | $-251$ | $-204$ | $-217$ | $-224$
Cu-Al | $-292$ | $-304$ | $-296$ | $-154$ | $-225$ | $-282$ | $-254$ | $-248$ | $-299$

###  II.2 Model calculations

The unary LJ crystal is a simple model to describe the stability of elemental
metals in the 2D and 3D structures ono_ito , which motivates us to apply the
BLJ model to interpret the stability of binary crystals $X$-$Y$. The BLJ model
has been used to study the glassy materials and viscous liquids KA , and
several ordered phases have been found at zero temperature wales2001 ;
fernandez2003 ; KA2018  such as coexisting phases of $X$ in the fcc structure
and $XY$ in the B2 (CsCl-type) structure KA2018 . In the present work, we
assumed that both types of particles have the same mass and interact via the
pair potential

$\displaystyle v_{\alpha\beta}(r_{ij})$ $\displaystyle=$ $\displaystyle
4\varepsilon_{\alpha\beta}\left[\left(\frac{\sigma_{\alpha\beta}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{\alpha\beta}}{r_{ij}}\right)^{6}\right],$
(2)

where $\alpha$ and $\beta$ are the atomic species, $X$ or $Y$, and $r_{ij}$ is
the interatomic distance between the atoms $i$ and $j$. In the present work,
the BLJ crystals are assumed to have the PHX structure and contain 16 atoms in
the unit cell. We used the Newton’s method to optimize the lattice parameter
of the hexagonal cell, and used the Broyden-Fletcher-Goldfarb-Shanno algorithm
numerical_recipe  to relax the positions of 16 atoms. The Eq. (1) is used to
calculate the formation energy in the BLJ crystals.

The pair potential in Eq. (2) has the minimum value of
$-\varepsilon_{\alpha\beta}$ at $r=\sigma_{\alpha\beta}2^{1/6}$, so that the
larger the $\sigma_{\alpha\alpha}$, the larger the lattice constant of the PHX
structure. We regard $\varepsilon_{XX}$ and $\sigma_{XX}$ as the energy units
and the length units, respectively, and chose $\varepsilon_{XX}=1$ and
$\sigma_{XX}=1$. The model parameters in the BLJ model are the
$\varepsilon_{XY}$, $\varepsilon_{YY}$, $\sigma_{XY}$, and $\sigma_{YY}$. The
average value of the interaction length was assumed, i.e.,
$\sigma_{XY}=(\sigma_{XX}+\sigma_{YY})/2$. In addition, we truncated and
shifted the pair potential at a cutoff distance of $2.5\sigma_{\alpha\beta}$
as in Ref. KA .

##  III Results and Discussion

###  III.1 Cu-Au

To determine the lowest formation energy structure for 2D Cu-Au, we first
studied the five models shown in Fig. 1: straight or zigzag chains of atoms
$X$ and $Y$ are aligned alternately in the first three models, while thicker
chains and clustered atoms are observed in the next two models. This is
motivated by the analogy between 2D and 3D structures: when the L10 CuAu is
elongated along the $c$ axis to satisfy $c/a=\sqrt{2}$, one obtains fcc CuAu,
and the (111) surface consists of straight chains of Cu and Au atoms. The L10
structure is also constructed by stacking the square lattice of Cu and Au
along the $c$ axis alternately. We thus expect that the chain-like structures
would have the lowest formation energy structure for the Cu-Au systems. For
comparison, we also studied the 2D Cu-Ag and Ag-Au, where the Cu and Ag atoms
are known to be immiscible and form a separated phase, and the Ag and Au atoms
can be mixed but does not form ordered phases. In this sense, the Cu8Au8 and
Ag8Au8 would have the chain or zigzag structures, while the immiscible Cu-Ag
systems prefer the stripe or diamond structures in Fig. 1. Table 1 lists the
values of $\Delta E_{j}$ for the five models, where the negative values of
$\Delta E_{j}$ indicate that mixing different atoms is more preferable in
energy. For the Cu8Au8 and Ag8Au8, the chain, zigzag, and zigzag2 structures
are more stable than the stripe and diamond structures, while for the Cu8Ag8,
an opposite tendency is observed, as expected.

Figure 2: Running the BO on the PHX Cu-Au with different three random seeds.
The BO arrives at the minimum value of $\Delta E_{j}$ (horizontal dashed) in
the 86 steps (blue solid) and 125 steps (green solid) by assuming different
random seeds. The other BO (red dashed) predicts the Cu10Zn6-type structure,
shown in Fig. 4.

It is desirable to avoid the use of biased models in Fig. 1. We thus searched
for the configuration showing the lowest $\Delta E_{j}$ by using the BO, and
found the optimized configuration having the chemical formula of Cu9Au7
showing $\Delta E_{j}=-78$ meV/atom. As shown in Fig. 2, the lowest $\Delta
E_{j}$ is obtained at about 100 steps in the BO runs. In the optimized
configuration (see Fig. 1), the Cu-based hexagon ring containing one Au atom
forms a large hexagonal lattice, into which upward and downward triangles of
Au atoms are embedded. Among six nearest neighbor atoms, each atom has
different distribution of species: an Au atom is surrounded by four or six Cu
atoms, and an Cu atom is surrounded by two or three Au atoms. This is in
contrast to the chain, zigzag, and zigzag2 structures, where all Au and Cu
atoms are surrounded by four Cu and Au atoms, respectively, and it may be
difficult to predict the optimized configuration from the analogy to the L10
structure.

Figure 3 shows the phase diagram for the 2D Cu-Au that was constructed by 7279
configuration calculations. The minimum of the $\Delta E_{j}$ is observed when
the Au concentration is $7/16=0.4375$, which confirms the validity of the
prediction based on the BO. It should be noted that the effect of structure
relaxation is strong enough to change the value of $\Delta E_{j}$ from
positive to negative value and to change the formation energy alignment
between different structures. For the unrelaxed structure, the Cu9Au7-type
structure has $\Delta E_{j}=-42$ meV, while the Cu10Zn6-type structure (see
Fig. 4 below) shows the lowest value of $\Delta E_{j}=-55$ meV. The structure
of the relaxed Cu9Au7 is depicted in Fig. 1, where the size of the Cu
hexagonal ring is contracted, while that of the Au triangle is expanded.

Figure 3: The $\Delta E_{j}$ for the Cu-Au system. The values of the unrelaxed
and relaxed structures are indicated by blue and red circles, respectively.
Figure 4: The optimized configuration of the Cu-$X$ systems on the hexagonal
plane for $X=$ Al, Be, Ga, Pd, Pt, and Zn. The blue ball indicates the Cu
atom, and the solid line (red) indicates the unit cell.

###  III.2 Cu-$X$

Having established the relevance of the BO in the search for the lowest
$\Delta E_{j}$ of 2D Cu-Au, we apply the same approach to explore other
ordered alloys. In the present work, we focus on the Cu-$X$ systems with $X$
being a metallic element in the periodic table. We first calculated the
$\Delta E_{j}$ in Eq. (1) for the zigzag and stripe structures in Fig. 1, and
selected the Cu-$X$ systems having negative $\Delta E_{j}$. We found that the
$X$ includes Al, Au, Ba, Be, Ca, Ga, Hf, In, Li, Lu, Mg, Pd, Pt, Sc, Sn, Sr,
Ti, Y, and Zn. However, when $X$ is the alkali earth metals (Ca, Sr, and Ba)
and the group 4 metals (Ti and Hf), the $\Delta E_{j}$ of the stripe structure
is lower than that of the zigzag structure. This implies that the energy gain
is small when the Cu and $X$ atoms are alloyed, so that we exclude these $X$
below.

We next applied the BO approach to search for the optimized mixing between the
Cu and $X$ atoms in the $4\times 4$ sites on the hexagonal plane. When the
atomic size mismatch is large between the Cu and $X$ atoms, many steps will be
needed in the structure relaxation within the DFT. In the present study, we
studied the atom $X$ satisfying the condition that the relative error of the
lattice constant between the 2D Cu ($2.428\times 4$ Å) and 2D $X$ in the PHX
structure is less than 10 percent. We thus studied the cases of $X=$ Al, Be,
Ga, Pd, Pt, and Zn, in addition to $X=$ Au.

Figure 4 shows the optimized configurations of the Cu-$X$ on the hexagonal
plane for the $X$ selected above. For the Cu-Be system, the Be-rich structure,
Cu7Be9, has the lowest value of $\Delta E_{j}$, and is isostructural with the
Cu9Au7 found by the BO, with Cu on the Au site and Be on the Cu site. This is
due to the similar value of the relative difference of the lattice constant
for the elemental metals in the PHX structure: The value of $\Delta a=|a_{\rm
PHX}({\rm Cu})-a_{\rm PHX}(X)|/a_{\rm PHX}({\rm Cu})$ is estimated to be 13.2
% for $X=$ Au and 12.4 % for $X=$ Be, where $a_{\rm PHX}({\rm Au})=2.748\times
4$ Å and $a_{\rm PHX}({\rm Be})=2.126\times 4$ Å were assumed. For the Cu-Zn
and Cu-Pd systems, there are elongated hexagons containing the Zn and Pd
dimers, where the three elongated hexagons share a triangle of the Cu atoms.
As mentioned, this structure has the lowest value of $\Delta E_{j}$ for the
unrelaxed Cu-Au system. In the running of the BO for the Cu-Ga, Cu-Pt, and Cu-
Al systems, the chain or zigzag structures have the lowest value of $\Delta
E_{j}$. The values of $\Delta E_{j}(X_{n}Y_{m})$ for several $j$s are listed
in Table 1. It should be noted that weakly-bonded systems ($-100\lesssim\Delta
E_{j}<0$ meV) tend to show the formula of $X_{n}Y_{m}$ with $n\neq m$. This
assumption will be rationalized below by using the BLJ model.

Nepal et al. have proposed that the $X$-$Y$ system is weakly-bonded when $X$
and $Y$ have completely filled $d$-bands, by assuming the B2 structure and
performing DFT calculations ruzsinszky2019 ; ruzsinszky2020 . Such
combinations of $X$ and $Y$ include the Cu-Au, Cu-Zn and Cu-Pd systems, which
are consistent with the present findings. In addition, they also proposed that
the Ag-Zn, Ag-Cd, Au-Cd, and Au-Zn are weakly-bonded systems.

In respect to the composition ratio, the analogy between 2D and 3D structures
cannot hold for Cu-Zn, and Cu-Pd systems. By referring to the Materials
Project database materialsproject , the Zn-rich phase of Cu17Zn35 and the CuPd
in the B2 structure have the lowest formation energy for Cu-Zn and Cu-Pd
systems, respectively. On the other hand, the Be-rich phase was found to be
the most stable structure in both 2D and 3D Cu-Be systems, where the latter is
identified as CuBe2 in the C15 (Laves) structure.

Figure 5: The lowest energy structures for the BLJ crystal in the PHX
structure when the pair potential parameters of $\varepsilon_{XY}$,
$\varepsilon_{YY}$, and $\sigma_{YY}$ are tuned. The cutoff radii for the
interatomic interaction is set to be $r_{\rm cut}=2.5\sigma_{\alpha\beta}$
except for the lower right. “PS” stands for the phase separation, and the
hyphen in $(\varepsilon_{XY},\varepsilon_{YY},\sigma_{YY})=(1,1,1)$ indicates
the unary LJ crystal.  Figure 6: The $\varepsilon_{XY}$-dependence of $\Delta
E_{j}$ for the BLJ crystals for nine structures, when $\varepsilon_{YY}=2.5$
and $\sigma_{YY}=1.2$ (see the lower middle panel in Fig. 5). The calculated
data for the $X_{9}Y_{7}$ and zigzag2 structures are also shown by circles.
The lowest energy structure changes from the stripe to $X_{9}Y_{7}$ to zigzag2
across the vertical lines. The horizontal dashed indicates $\Delta E_{j}=0$.

###  III.3 Binary Lennard-Jones

Without prior knowledge on the configurations predicted by the BO, it may be
difficult to find the Cu9Au7-type and Cu10Zn6-type structures (in Fig. 4)
because the condition $n\neq m$ in Cu${}_{n}X_{m}$ holds. To understand when
the structures with $n\neq m$ are more stable than those with $n=m$, we
calculated the $\Delta E_{j}$ for the BLJ crystals, as described in Sec. II.2.
We studied the $X_{8}Y_{8}$ in the first five structures as shown in Fig. 1,
the $X_{9}Y_{7}$ (Cu9Au7-type) structure (see Fig. 1), the $X_{10}Y_{6}$
(Cu10Zn6-type) structure (see Fig. 4), and the isostructural configurations
(i.e., $X_{7}Y_{9}$ and $X_{6}Y_{10}$). Figure 5 shows the lowest formation
energy structures as a function of $\varepsilon_{XY}$ and $\varepsilon_{YY}$
for several $\sigma_{YY}$. When $\varepsilon_{YY}\gg\varepsilon_{XY}$, the
phase separation (i.e., $\Delta E_{j}>0$) is preferred because the interatomic
interaction between the $X$ and $Y$ atoms is not strong enough to alloy with
different species. When $\varepsilon_{XY}$ is increased, the $X$ and $Y$ atoms
are mixed, forming an ordered structure that depends on the parameters of
$(\sigma_{YY},\varepsilon_{XY},\varepsilon_{YY})$: When $\sigma_{YY}=1.0$, the
chain structure is the lowest formation energy structure for
$\varepsilon_{YY}<\varepsilon_{XY}$; when $\sigma_{YY}=0.9$ and 1.0, the
zigzag structure appears around the boundary
$\varepsilon_{YY}\simeq\varepsilon_{XY}$. In addition, another zigzag
structure with a longer period (zigzag2) appears for large $\varepsilon_{XY}$;
and when $\sigma_{YY}=0.8$ and 1.2, such a long-period zigzag structure
becomes the lowest formation energy phase for
$\varepsilon_{YY}<\varepsilon_{XY}$. We also studied the cases with the cutoff
radius of $3.0\sigma_{\alpha\beta}$. However, the phase diagrams are basically
the same as those calculated by using $2.5\sigma_{\alpha\beta}$, while the
zigzag structure overcomes the zigzag2 structure for large $\varepsilon_{XY}$.

It is noteworthy that the Cu10Zn6-type and the Cu9Au7-type structures can be
found when $(\sigma_{YY},\varepsilon_{XY},\varepsilon_{YY})=(0.8,2.5,3.0)$ and
$(\sigma_{YY},\varepsilon_{XY},\varepsilon_{YY})=(1.2,2.0,2.5)$, respectively.
Although the inequality $\varepsilon_{YY}>\varepsilon_{XY}$ holds, moderately
large value of $\varepsilon_{XY}$ barely prevents the $X$-$Y$ system from the
phase separation. To study the impact of $\varepsilon_{XY}$ on the energy
alignment, we plotted the values of $\Delta E_{j}$ for several structures in
Fig. 6 by assuming $(\varepsilon_{YY},\sigma_{YY})=(2.5,1.2)$. As
$\varepsilon_{XY}$ increases, $\Delta E_{j}$ decreases, and becomes negative
value around $\varepsilon_{XY}=1.8\sim 2.2$ depending on the structure. This
shows that in the limit of small and large $\varepsilon_{XY}$ the $X$-$Y$
system can be regarded as an immiscible and a strongly-bonded system,
respectively, and for moderate $\varepsilon_{XY}$ the $X$-$Y$ system is a
weakly-bonded system. Among nine models, the lowest formation energy structure
changes from the stripe to $X_{9}Y_{7}$ (at $\varepsilon_{XY}=1.3$) to zigzag2
(at $\varepsilon_{XY}=2.1$) structures with $\varepsilon_{XY}$. The difference
of $\Delta E_{j}$ between the $X_{9}Y_{7}$ and $X_{10}Y_{6}$ structures are
small (less than 0.001 for $\varepsilon_{XY}\geq 1.4$). In this respect, the
weakly-bonded systems prefer the Cu9Au7-type and the Cu10Zn6-type structures.
Furthermore, the BLJ model calculations are consistent with the small values
of $\Delta E_{j}$ in the Cu-$X$ with $X=$ Au, Be, Pd, and Zn (larger than
$-100$ meV/atom listed in Table 1).

##  IV Conclusion

We have demonstrated that the combined use of DFT and BO approach enables us
to predict the lowest formation energy structure of the Cu-Au system in the
PHX structure with 16 atomic sites. The stable structure has the chemical
formula of Cu9Au7, which is a counterintuitive distribution pattern of Cu and
Au atoms because no structural similarities with the L10 CuAu can be found. We
also explored the stable structures of Cu${}_{n}X_{m}$ with $n+m=16$ and $X=$
Be, Al, Zn, Ga, Pd, and Pt, in which the lattice constant of $X$ in the PHX
structure is similar to that of Cu. The lowest formation energy structure
satisfies the condition $n\neq m$ when $X=$ Be, Pd, and Zn. The BLJ model
calculations suggest that such structures can appear for weakly-bonded systems
located in between the phase-separated and strongly-bonded systems. It is
desirable to explore the stable structures for multilayered systems, leading
to a fundamental understanding of the structural similarity between 2D and 3D
systems. We expect that many interesting structures will be found when the
boundary condition (i.e., the PHX structure containing 16 atoms) is removed,
or when other $X$-$Y$ systems are considered.

The present approach can be extended to investigate the stable spin
configurations of magnetic materials in the PHX structure, which will enable
us to find the lowest energy anti-ferromagnetic phase that overcomes the
stability of stripe order in 2D elemental metals ren . It is interesting to
determine the noncollinear spin configuration of 2D Mn (i.e., including spin-
orbit coupling) because the 3D Mn has a cubic cell containing 58 atoms,
showing a complex anti-ferromagnetic ordering in the ground state Mn2020 .

In the present work, we assumed the planar structure, so that the 2D Cu-$X$
might be unstable to the out-of-plane vibrations called the flexural phonon
modes. We expect that these systems may be stabilized by using monolayer
and/or bilayer graphene: For example, one-forth of the lattice constant of the
optimized Cu9Au7 ($\simeq 2.58$ Å) is close to the lattice constant of
graphene (2.46 Å), so that the Cu9Au7 can be realized on the graphene
substrate, as in the experimental synthesis of buckled honeycomb CuAu (2.68 Å
zagler ). Another possibility is the use of the 2D nanospace realized by the
gap of the bilayer graphene kanetani ; ago , where the effect of the van der
Waals interaction forces between the 2D systems and the graphene may be
important to stabilize the out-of-plane vibrations.

###### Acknowledgements.

This work was supported by JSPS KAKENHI (Grant No. JP21K04628). The
computation was carried out using the facilities of the Supercomputer Center,
the Institute for Solid State Physics, the University of Tokyo, and using the
supercomputer “Flow” at Information Technology Center, Nagoya University.

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