# Comparing Importance Sampling Based Methods for Mitigating the Effect of
Class Imbalance

Indu Panigrahi and Richard Zhu11footnotemark: 1
Princeton University
{indup<EMAIL_ADDRESS>Equal contribution.

###### Abstract

Most state-of-the-art computer vision models heavily depend on data. However,
many datasets exhibit extreme class imbalance which has been shown to
negatively impact model performance. Among the training-time and data-
generation solutions that have been explored, one subset that leverages
existing data is importance sampling. A significant portion of this work
focuses primarily on the CIFAR-10 and CIFAR-100 datasets which fail to be
representative of the scale, composition, and complexity of current state-of-
the-art datasets. In this work, we explore and compare three techniques that
derive from importance sampling: loss reweighting, undersampling, and
oversampling. Specifically, we compare the effect of these techniques on the
performance of two encoders on an impactful satellite imagery dataset,
Planet’s Amazon Rainforest dataset, due to its challenging nature and existing
class imbalance. Furthermore, we perform supplemental experimentation on a
scene classification dataset, ADE20K, to test on a contrasting domain and
clarify our results. Across both types of encoders, we find that up-weighting
the loss for and undersampling has a negigible effect on the performance on
underrepresented classes. Additionally, our results suggest oversampling
generally improves performance for the same underrepresented classes.
Interestingly, our findings also indicate that there may exist some redundancy
in data in the Planet dataset. Our work aims to provide a foundation for
further work on the Planet dataset and similar domain-specific datasets. We
open-source our code for future work on other satellite imagery datasets as
well.

##  1 Introduction

Detecting illegal slash-and-burn and mining activities in protected
rainforests is crucial for ensuring the well-being of such areas, and
satellite imagery can facilitate this process. Satellite imagery can cover
vast areas, so there can exist a large the amount of data to parse.
Classification models can expedite this process by automating the
categorization of satellite images according to their captured patterns.

However, many large and widely-used datasets exhibit a long-tail distributions
where some classes have significantly more examples than others (Figure 1)
[62, 32, 14, 63, 61, 17]. Oftentimes, machine learning models achieve higher
accuracies on overrepresented classes than on underrepresented classes,
similar to how a human would be better at classifying images corresponding to
classes for which they have seen more examples [3, 22, 46, 19]. However, this
induced bias can lead to dangerous and unfair model predictions [4, 25, 53,
54].

We explore a satellite imagery dataset, the Planet Rainforest dataset [11,
42], that demonstrates this trend. In fact, the detrimental activities that
users would want to detect are some of the worst-represented classes in the
dataset (Figure 1). Additionally, this type of distribution becomes more
difficult to remediate when each image has multiple labels, as is the case
with the Planet dataset.

Figure 1: Long-tail distribution of classes in the Planet dataset. The
frequency of images across the $17$ labels in the Planet dataset form a skewed
distribution where some labels, such as “primary” and “agriculture”, have many
images and other labels, such as “blooming” and “blow_down”, have very few
images. Important labels, such as “slash_and_burn” and “artisanal_mining”, are
some of the most underrepresented classes in the dataset.

From a theoretical machine learning perspective, in addition to bounding
generalization error as we evaluate models on test sets, we may also be
interested in developing empirical methods for minimizing expected error. By
understanding the causes of poor generalization in practice, we can gain
intuition in order to further constrain bounds when developing models in
practice.

A natural solution to this issue of class imbalance would be to simply collect
or generate more data. Depending on the application, collecting data is often
time-consuming and costly, and generating data may not produce sufficiently-
realistic representations. In general, exploring ways to leverage existing
data would be beneficial.

Importance sampling covers a subset of methods that recycles existing data to
mitigate the effects of class imbalance [5, 18, 23, 43, 26]. To that end, we
explore and compare three importance sampling based methods: loss reweighting,
undersampling, and oversampling.

Specifically, we experiment with two datasets from different domains, the
Planet Rainforest [11, 42] and the ADE20K [61] datasets, and with two image
encoders, CLIP [39] and ResNet-18 [15]. We primarily focus on the Planet
dataset since we believe that it provides varied, challenging tasks that can
accentuate the effects of class imbalance. However, we include supplemental
experimentation on ADE20K, for completeness and to observe how generalizable
our findings are. We strive to evaluate our findings on datasets that test a
skillset representative of broader model capabilities.

Across the two types of encoders, we find that up-weighting the loss on
underrepresented classes has a negligible effect on performance. Additionally,
we find that oversampling appears to be more effective than undersampling.
That being said, our results suggest that undersampling could still be a
valuable method if we mindfully select samples rather than at random. Lastly,
our experiments indicate that there may exist some redundancy in the Planet
dataset which motivates new avenues for future work.

##  2 Related Work

#### Class Imbalance Problem.

Class imbalance occurs when some classes have significantly more examples than
others in a dataset. Imbalanced distributions naturally arise across many
types of datasets ranging from medical imaging to environmental satellite
imagery [28, 41, 57, 16]. While other forms of imbalances exist (e.g., object-
level for object detection and pixel-level for segmentation), our work
explores solutions for class imbalance since we examine image classification.
While the class imbalance problem has been well-studied in traditional non-
deep machine learning and computer vision, recent studies have noted the lack
of work on studying the effects of and methods for counteracting class
imbalance across a variety of datasets and imbalance levels [22]. We work with
a satellite imagery and a scene classification dataset to evaluate on
different domains. Prior work has shown class imbalance to deteriorate model
performance on underrepresented classes [52, 33, 12, 38]. However, these
classes can be extremely important, as is the case with the Planet dataset
(Figure 1).

#### Data generation.

A natural solution to the class imbalance problem would be to collect more
data to balance the dataset. Since manual data collection can be quite arduous
for specialized applications (e.g., medicine, botany, and geosciences) [2, 47,
55, 17], an alternative is to generate additional data via self-supervised
learning [60, 20], semi-supervised learning [30, 58], or generative
adversarial networks (GANs) [7, 13]. However, one common limitation is that
the generated data can sometimes be low in quality, particularly for
specialized applications such as for those represented in this paper (e.g.
satellite imagery of rainforests). Furthermore, leveraging existing data would
still be data-efficient but simpler. As a result, we choose to focus on
comparing methods that leverage existing data.

#### Cost-sensitive learning.

One such method is cost-sensitive learning. Cost-sensitive learning is a
progressive training method where higher-cost examples contribute more
significantly in updating the model. In practice, this means the user can
tailor the model to heavily penalize certain misclassifications [9, 59]. For
example, if the model performs poorly on an underrepresented class, the user
can increase the penalty for mistakes on that class by influencing the cost
(i.e., loss). In this work, we implement a loss reweighting scheme which we
describe in Section 5.

#### Importance Sampling.

Another related subset of methods that overlaps with cost-sensitive learning
is importance sampling. Importance sampling is useful when the test
distribution $q$ is not accurately reflected in the training distribution $p$
[18, 23, 43, 26]. For some example $x$, if we are estimating some function
$f(x)$, importance sampling weights $x$ based on its likelihood ratio
$\frac{q(x)}{p(x)}$ as shown in Equation 1 [5].

$\displaystyle\mathbb{E}_{p}\left[\frac{q(x)}{p(x)}f(x)\right]$
$\displaystyle=\int_{x}\frac{q(x)}{p(x)}f(x)p(x)dx$ (1)
$\displaystyle=\int_{x}q(x)f(x)dx$ (2) $\displaystyle=\mathbb{E}_{q}[f(x)]$
(3)

The idea of importance sampling forms the basis for the three methods that we
explore which we describe in the next section.

##  3 Methods

Our work explores and compares loss reweighting and two types of resampling:
undersampling and oversampling. In this section, we provide a description of
each method.

#### Reweighting loss.

This technique reweights the loss at training time to amplify the loss for
low-frequency classes and/or diminish the loss for other classes. One approach
scales the loss of all low-frequency classes by a constant (Figure 2), while
another scales the loss of all classes relative to their frequency. We choose
to do the former because we find that it offers more granularity in
controlling which classes are amplified in situations where two labels may be
very similar. For example, if we have an image classification task where the
classes ”roadway” and ”highway” appear moderately frequently, we may still
wish to consider these classes as low-frequency classes to improve the model’s
ability to distinguish between the two (likely very similar) images.

Figure 2: Reweighting Loss Method. Our approach scales the loss $\ell(x,y)$
where $x$ is the image and $y$ is the label by a constant factor $k$ if the
instance belongs to a low-frequency class. We evaluate with a range of values
for $k$ (Section 5), specifically $1.5$, $2$, $2.5$, $3$, $5$, and $10$.

Prior work has previously implemented variants of the latter approach. One
such work evaluates the effectiveness of learning a mapping from classes to
reweighting factors [50]. Using an MLP with a single hidden layer, they
observe some performance gains on the CIFAR-10 and -100 datasets, achieving
test accuracies on ResNet-32 that are greater than or equal to those on a
class-balanced version of the dataset. Another work simply scales the loss by
the inverse class frequency. Their method, class-balanced focal loss, achieves
a nominally better training loss on the ILSVRC 2012 [44] and iNaturalist 2018
[17] datasets after sufficiently many epochs, while results at earlier epochs
underperform the default softmax cross-entropy [8]. On CIFAR-10 and -100, they
report gains, albeit marginal ones. We include a discussion about another
related work that takes the latter approach in the context of our results for
completeness (Section 5) [5].

#### Undersampling.

Undersampling reduces the number of examples in overrepresented classes by
randomly sampling a subset of a predefined size (Figure 3(a)). This size is
usually the size of the lowest-frequency class [3, 31]. This method
effectively decreases the total number of elements. As a result, the number of
examples seen from each class during training is generally more uniform, so
intuitively, the model is expected to perform more uniformly across classes.
One of the first papers that discussed class imbalance highlighted the
potential performance improvements of undersampling methods [27].

Since undersampling removes examples from the training set, this method
appears to be more suitable for large datasets where we can afford to prune
the dataset without significantly affecting results [3]. While the exact
threshold for such a decision seems nebulous, most prior work finds that the
thresholds that we use (typically undersampling all elements to a threshold
approximately equal to the number of elements in the least populous class
being trained on), does not significantly decrease performance [3]. Thus,
applying undersampling to the Planet dataset seems reasonable.

(a) Undersampling.

(b) Oversampling.

Figure 3: Types of Resampling. Oversampling (b) trains on repeated elements
from underrepresented classes, while undersampling (a) does the opposite,
training on a randomly-selected subset of elements from overrepresented
classes. The threshold chosen for the number of images to resample or remove
is dataset-dependent.

#### Oversampling.

Similar to undersampling, oversampling aims to create a more uniform
distribution of training examples across classes. However, this process
instead repeats training examples from low-frequency classes via random
sampling until those classes reach a comparable frequency to the other classes
(Figure 3(b)). Both undersampling and oversampling can be tricky for multi-
label classification (e.g. with the Planet dataset). There exist several
approaches to ensure a roughly similar number of elements in both resampling
methods, and we discuss our approach in the next section (Section 4).

##  4 Experimental Details

#### Data and Tasks.

We primarily experiment with the Planet Rainforest dataset (Figure 5) and
include supplemental experimentation on the ADE20K dataset (Figure 6). The
Planet dataset contains satellite imagery of the Amazon Basin in South
America, and each image has 4 bands (red, green, blue, and near-infrared).
This dataset consists of fine-grain categories such as “conventional mining”
and “artisanal mining”. Each image includes a label that indicates cloud
coverage (e.g., “clear”, “partly_cloudy”) along with labels that describe the
areas in the images (Figure 5). As a result, each image in the Planet dataset
typically has at least two labels.

Figure 4: Distribution of classes in ADE20K.

This dataset demonstrates a long-tail distribution, and the classes that
indicate illegal or dangerous forest activites (e.g., “slash_and_burn” and
“artisanal mining” are among the lowest-frequency classes (Figure 1). This
distribution is problematic because machine learning models more generally
tend to perform poorly on underrepresented classes.

Figure 5: Sample Images from the Planet dataset. Figure from Planet’s Kaggle
website [11]. The Planet dataset consists of fine-grain categories such as
“conventional mining” and “artisanal mining”. Each image has multiple labels
one of which is a cloud cover label, such as “partly cloudy” or “clear”.

The ADE20K dataset contains images of commonplace scenes and has a few levels
of annotations. There are scene classification labels where each image maps to
one label and segmentation annotations for the individual objects in each
image. In this work, we focus on image classification, so we use only the
scene classification labels. The ADE20K dataset also demonstrates a long-tail
distribution.

Figure 6: Sample Images from the ADE20K dataset. The ADE20K dataset is a scene
classification dataset with one label per image. ADE20K includes object-level
and segmentation labels as shown; however, we only use the image
classification labels. Figure from the ADE20K website [61].

We choose the Planet dataset because of the challenging skillset involved and
class imbalance, consequently testing multi-label classification. We include
ADE20K in our reweighting experiments to disentangle the results by testing
single-label classification (Section 5) and provide a contrasting domain.
However, the primary focus of this work is on the Planet dataset.

We use all $17$ classes from Planet and select the $20$ most frequent classes
(out of the $150$ total) from ADE20K. Both datasets exhibit long-tail
distributions with Planet having the more severe of the two (Figure 1, 4).
Since these are challenge datasets, the test sets are not publicly available,
so we apply a $60$-$20$-$20\%$ split to the training and validation sets
combined to produce our training, validation, and test sets. In our results,
we report the test accuracies.

#### Define low-frequency classes.

Prior works mostly tailor the definition of low-frequency classes to their
datasets by selecting classes based on certain features or by predefining a
frequency threshold [21, 37]. A more general option is to define the low
frequency classes to be those that fall one standard deviation below the mean.
However, for Planet, the mean frequency is $5464.82$, and the standard
deviation is $8214.73$ which is too large to follow this standard. Instead, we
define the low-frequency classes to be those that have frequencies that fall
within the $25^{th}$ percentile of all the frequencies. To maintain
consistency, we use the same definition to select the low-frequency classes in
ADE20K.

#### Resampling.

For undersampling, we choose a threshold of $200$ for Planet because this is
approximately the frequency of the lower-frequency classes. Thus, we randomly
sample from classes with frequencies that exceed this threshold to reduce
those classes to the threshold.

For oversampling, we choose a threshold of $30,000$ for Planet because this is
approximately the frequency of the higher-frequency classes. Similar to
undersampling, we randomly sample from classes with frequencies lower than
this threshold until they reach the threshold. However, before doing so, we
ensure that all the images from those classes are included so that each image
appears at least once.

(a) Undersampled distribution of the Planet dataset.

(b) Oversampled distribution of the Planet dataset.

Figure 7: Resampled distributions of the Planet dataset.

Since the images in Planet can have multiple labels, ensuring a perfectly
uniform distribution is unreasonable. In particular, the ”primary” label
indicates the presence of trees, and ”clear” indicates that there exists no
cloud cover. As a result, these two labels overlap with many images,
especially as each image has a cloud cover label (i.e., ”clear”,
”partly_cloudy”, ”haze”). However, we achieve a generally uniform distribution
among the more disjoint classes (Figure 7) which is significantly less skewed
than the original distribution (Figure 1).

#### Architecture.

Our architecture consists of a pretrained encoder followed by a simple
classifier (Figure 8). We experiment with two encoders pretrained on ImageNet:
CLIP [39] and ResNet-18 [15]. We implement a simple classifier in PyTorch that
consists of the following layers: Linear, ReLU, Linear, and Sigmoid. We use a
binary cross-entropy loss function. The encoders are frozen, so we
specifically train the classifier.

Figure 8: Model Architecture. We implement a simple classifer that takes as
input embeddings of the input images generated by a frozen, pretrained
encoder. We experiment with CLIP [39] and ResNet-18 [15] as encoders.

#### Training.

After performing a hyperparameter search, we decide to train each model for
$20$ epochs using a learning rate of $1e-3$, batch size of $64$, and dropout
rate of $0.2$. We train all the models with the same hyperparameters to
control for variance in performance caused by hyperparameters as do prior
works.

##  5 Results & Discussion

We make 3 key observations. First, that upweighting has a small effect on
class performance. While we observe this trend in experimental settings where
we exclusively up-weight low-frequency classes, and leave mid- and high-
frequency classes unchanged, we believe that a similar trend would exist even
if all classes were oversampled to a constant frequency. This conclusion is
supported by the largely similar proportional sampling techniques from prior
works (see Section ) that show only nominal performance gains. Second, we find
that undersampling has a similarly small effect on class performance and even
(albeit somewhat unsurprisingly since parts of the datasets are being omitted
from the model’s training data) reduces performance on mid- and high-frequency
classes. We lastly observe, a similar trend in oversampling. Most
surprisingly, we note that the traditionally well-accepted and intuitive
methods of reducing the effect of dataset bias appear to be largely
ineffective with current or recent state of the art models.

In order to investigate the generalizability of these results, we consider
performance across two different datasets (Planet and ADE20K) which have
drastically different features. Their label distributions differ (see Figure
7), and their contrast, color distribution, and purpose vary (see Figure 5 and
6). Though we are primarily interested in dataset characteristics, we also
attempt to study the generalizability of our finding with respect to the
models themselves. We find our conclusions to hold fairly well between both
CLIP and ResNet-18 models.

###  5.1 Up-weighting the loss for low-frequency classes does not have a
significant effect on the performance on any of the classes.

(a) Up-weighting accuracy with CLIP encoder.

(b) Up-weighting accuracy with ResNet-18 encoder.

Figure 9: Up-weighting accuracy on the Planet dataset on low-frequency
classes. These graphs depict the accuracies when scaling the loss for low-
frequency classes by factors of $1.5,2,2.5,3,5,$ and $10$ with different
encoders; the factor of $1$ corresponds to the baseline accuracy. Both panels
depict the accuracies on the low-frequency classes only; see Figure 10 for the
remaining classes. For both encoders, we find that there exists no consistent
trend across scaling factors or within a factor.

(a) Up-weighting accuracy on the remaining classes in Planet when using CLIP.

(b) Up-weighting accuracy on the remaining classes in Planet when using
ResNet-18.

Figure 10: Up-weighting accuracy on Planet on remaining classes. These graphs
depict the accuracies on the remaining classes in Planet when scaling the loss
for low-frequency classes by factors of $1.5,2,2.5,3,5,$ and $10$; the factor
of $1$ corresponds to the baseline accuracy. Panel (a) corresponds to using
CLIP and panel (b) corresponds to using ResNet-18.

Across both models, we find that the performance does not exhibit a consistent
improvement or decay when we up-weight the loss of the low-frequency classes
in Planet (Figure 9, 10). The results for multi-label classification should be
interpreted with the caveat that each image can have more than one label. For
example, if an image has the labels ”primary” and ”slash_and_burn”, then we
would up-weight the loss on the image because ”slash_and_burn” is a low-
frequency class. However, in doing so, we implicitly up-weight the loss on
”primary”. Thus, the up-weighting is not cleanly done on one class and can
cause sporadic drops or gains in performance. To disentangle these effects, we
check the performance of the same up-weighting scheme on single-label
classification. However, we again find that the performance of the models does
not significantly change when we up-weight the loss of the low-frequency
classes (Figure 11).

Figure 11: Up-weighting accuracy on ADE20K with CLIP encoder on low-frequency
classes. This graph depicts the accuracies when using a CLIP encoder and
scaling the loss for low-frequency classes by factors of $1.5,2,2.5,3,5,$ and
$10$. The factor of $1$ corresponds to the baseline accuracy. Similar to the
results from the Planet dataset, we find that there is no consistent trend
across the scaling factors. Figure 12: Up-weighting accuracy on the remaining
classes in ADE20K when using CLIP. This graph depicts the accuracies when
using a CLIP encoder and scaling the loss for low-frequency classes by factors
of $1.5,2,2.5,3,5,$ and $10$. The factor of $1$ corresponds to the baseline
accuracy. Similar to the results from the Planet dataset, we find that there
is no consistent trend across the scaling factors.

For single-label classification, the performance on an up-weighted class
sometimes drops surprisingly (e.g., with the ”game_room” label in Figure 11).
However, a plausible explanation is that the training examples for that label
do not reflect the testing distribution, and the model learns meaningless
artifacts and/or makes a faulty association [35, 29, 34, 10]. For example, if
the training examples for the class ”game_room” tend to contain pool tables
and if the testing examples for ”game_room” do not contain pool tables, the
model could develop a dependence on the presence of pool tables to identify a
scene as ”game_room” and consequently misclassify the testing examples. This
hypothesis could be investigated further by comparing the saliency maps
generated for the training images to those generated for the testing images
[48, 51, 49].

In addition to our experiments, this finding is partly supported by prior
experimentation in Byrd & Lipton’s work on importance sampling [5]. Byrd &
Lipton use a reweighting scheme that weights each class according to its
relative frequency in the dataset. This scheme is slightly different from our
setup in that we use a constant scaling factor for all classes, regardless of
frequency. Nonetheless, Byrd & Lipton found that the impact of reweighting the
loss diminished over time, and our results are consistent with this finding.

###  5.2 Undersampling produces a comparable performance to the baseline on
the low-frequency classes while reducing the performance on the other classes.

Across both models, the performance when using undersampling roughly matches
or surpasses the baseline performance on low-frequency classes (Figure 13).
The situations where the performance improves compared to the baseline
performance are surprising because we use the same images for these classes.
One potential cause is stochasticity of training which could be resolved by
instead training multiple undersampling and baseline models and averaging the
performances assuming access to sufficient computational and storage
resources.

(a) Resampling accuracy on Planet’s low-frequency classes with CLIP encoder.

(b) Resampling accuracy on Planet’s low-frequency classes with ResNet-18
encoder.

Figure 13: Resampling accuracy on the low frequency classes in the Planet
dataset. The classes are sorted in descending order by frequency along the
x-axis.

The performance on the remaining classes often drops (e.g., as with the
”agriculture” and ”partly_cloudy” labels) (Figure 14). This trend seems
reasonable since undersampling removes data from these classes. However, it is
interesting to note that the performance on some of these classes remains
roughly the same (e.g., the ”selective_logging” and ”bare_ground” labels).
This trend suggests that the examples removed from those classes were
redundant with the examples that remained and that the examples that were
removed from the negatively-impacted classes did not contain redundant
information [56, 36, 1, 24]. Furthermore, we use approximately $21$ times less
data when undersampling, so the fact that the performance is still reasonable
across all classes is impressive and further supports the idea that there may
exist some redundancy in the data.

(a) Resampling accuracy on Planet’s remaining classes with CLIP encoder.

(b) Resampling accuracy on Planet’s remaining classes with ResNet-18 encoder.

Figure 14: Resampling accuracy on the remaining classes in the Planet dataset.
The classes are sorted in ascending order by frequency along the x-axis.

###  5.3 Oversampling generally improves performance across low-frequency
classes.

Among the low-frequency classes, oversampling generally matches or improves
upon the baseline performance (Figure 13). Since oversampling artificially
adds data by sampling from existing data, marginal improvements are reasonable
to expect. For situations where the performance is matched (e.g., ”blooming”
and ”blow_down” in Figure 13), the repetition of the data could have
outweighed any improvement. In particular, it is interesting to note that,
with ResNet-18, as the classes reduce in frequency (progress along the
x-axis), the gap between the baseline and oversampling performance reduces
(Figure 13(b)). This trend likely occurs because the fewer examples a class
contains, the more frequently those examples will be seen during training with
oversampling, so the model learns highly repetitive information about that
class.

Surprisingly, oversampling sometimes leads to a drop in performance in the
remaining classes, though to a lesser degree than undersampling (Figure 14).
This trend likely occurs because we stopped training prematurely. Oversampling
uses approximately $8$ times more data than the baseline, so using the same
number of epochs when training would not be sufficient. However, as noted in
Section 4, we train all models under the same hyperparameters to control for
comparison.

###  5.4 Comparison between methods

Surprisingly, up-weighting the loss on low-frequency classes, even by a
relatively large factor of $10$, does not produce consistent effects on the
performance of any of the classes. In fact, the two resampling methods seem to
produce more consistent effects than reweighting. Compared to undersampling,
oversampling appears to have the potential to cause the most improvement
provided more training time to account for the significant increase in data.
However, the performance when undersampling is surprisingly high considering
that we use $8$ times less data from Planet and perhaps selecting data
according to some criteria rather than at random would help.

##  6 Conclusion

In this work, we present a comparison of methods for mitigating the effect of
class imbalance that are based on the idea of importance sampling.
Specifically, we focus on a satellite imagery dataset and provide supplemental
experimentation with a scene classification dataset. Across two types of
encoders, CLIP and ResNet-18, we find that up-weighting the loss on low-
frequency classes has a negligible effect on the performance. Additionally, we
find that oversampling appears to be more effective than undersampling.
However, we also find that undersampling could still be a valuable method
though perhaps users should not select data at random.

In our discussion, we often noted that there likely exists some redundancy in
the data. This idea is part of the motivation for data distillation which
attempts to reduce a dataset to a few representative examples per class [40,
6, 45]. Thus, this could be an interesting alternative to undersampling via
random selection.

Additionally, we focus on bias due to underrepresentation. However, in our
discussion, we also hint about bias due to spurious correlation (i.e., when
the model forms a faulty association between certain image features and a
class) [54]. Thus, there exist other forms of bias to consider to develop
truly robust models, and as machine learning research heads towards a
direction of minimizing bias, training data plays an ever more important role
in shaping the representations formed through learned parameters.

Future work might also explore whether model accuracy improves sharply for a
label once that label’s training data size crosses above a certain threshold,
suggesting the existence of activation-like behavior at that cut-off.
Alternatively, the model accuracy might not become asymptotic — except as it
approaches 100% — in the quantity of data per label and instead accuracy could
increase continually. A study of this behavior across various leading models
could provide useful insights on the amount of data per class needed to
achieve certain performance expectations.

A macroscopic implication of these results is that, as a field, our
understanding of how black-box deep neural networks operate is still nascent.
As shown in this report, some of our intuitions on what should improve model
performance (e.g., “more data should improve model performance” and “up-
weighting loss should improve performance”) do not necessarily hold. As
machine learning rapidly expands to other fields, the importance of data
efficiency and model interpretability grows increasingly important both for
generalizability and for trusting AI systems.

#### Limitations.

In our experiments with up-weighting the loss, we scale the loss for each low-
frequency class by the same constant factor. Another option is to scale each
class by a factor that is proportional to the relative frequency of the class.
Then, among the low-frequency classes, the lowest-frequency class would be up-
weighted more than the more-frequent classes. However, Byrd & Lipton still
find that this does not improve performance [5].

Our model architectures used CLIP and ResNet-18 encoders pretrained on
ImageNet [44]. Satellite imagery comes from a starkly different domain than
ImageNet, so one potential improvement would be to train the encoders.
However, previous work has shown that pretraining is still beneficial, and we
did use the ADE20K dataset which comes from a similar domain as ImageNet.

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# Power Allocation for IRS-aided Two-way Decode-and-Forward Relay Wireless
Network

Xuehui Wang, Peng Zhang, Feng Shu, Weiping Shi, and Jiangzhou Wang,  _Fellow_
,  _IEEE_ This work was supported in part by the National Natural Science
Foundation of China (Nos. 62071234, 62071289, and 61972093), the Hainan Major
Projects (ZDKJ2021022), the Scientific Research Fund Project of Hainan
University under Grant KYQD(ZR)-21008, and the National Key R&D Program of
China under Grant 2018YFB180110 _(Corresponding authors: Feng Shu)_.Xuehui
Wang, Peng Zhang, and Feng Shu are with the School of Information and
Communication Engineering, Hainan University, Haikou, 570228, China.Weiping
Shi and Feng Shu are with the School of Electronic and Optical Engineering,
Nanjing University of Science and Technology, 210094, China.Jiangzhou Wang is
with the School of Engineering, University of Kent, Canterbury CT2 7NT, U.K.
Email: (e-mail: j.z.wang@kent.ac.uk).

###### Abstract

In this paper, an intelligent reflecting surface (IRS)-aided two-way decode-
and-forward (DF) relay wireless network is considered, where two users
exchange information via IRS and DF relay. To enhance the sum rate
performance, three power allocation (PA) strategies are proposed. Firstly, a
method of maximizing sum rate (Max-SR) is proposed to jointly optimize the PA
factors of user $U_{1}$, user $U_{2}$ and relay station (RS). To further
improve the sum rate performance, two high-performance schemes, namely
maximizing minimum sum rate (Max-Min-SR) and maximizing sum rate with rate
constraint (Max-SR-RC), are presented. Simulation results show that the
proposed three methods outperform the equal power allocation (EPA) method in
terms of sum rate performance. In particular, the highest performance gain
achieved by Max-SR-RC method is up to 45.2% over EPA. Furthermore, it is
verified that the total power and random shadow variable $X_{\sigma}$ have a
substantial impact on the sum rate performance.

###### Index Terms:

Intelligent reflecting surface, two-way decode-and-forward relay, power
allocation, sum rate.

##  I Introduction

There are many thorny problems in communication networks, such as the
propagation loss and multi path fading, which seriously deteriorate the
communication performance [1]. With the ability to intelligently adjust the
propagation environment, helpful multi path can be created by intelligent
reflecting surface (IRS) for coverage extension. IRS has become an emerging
technology with competitive advantages over the existing technologies [2]. IRS
consists of cost-effective, low-power, and passive reflecting units, which
reflect the signal independently to achieve passive beamforming for signal
enhancement, spectral and energy efficiency improvement [3, 4]. IRS has been
widely applied to different application scenarios, e.g., physical layer
security [5], simultaneous wireless information and power transfer (SWIPT)
[6], [7], mobile edge computing [8], and covert communication [9].

Due to the fact that IRS reflects signal with low-power consumption, which can
be regarded as a passive relay. While the conventional relay is an active
device with strong signal processing capability to amplify and forward (AF),
decode and forward (DF) signal. Its high hardware cost and high-power
consumption are inconsistent with our low-energy-consumption communication
demand.

How to make full use of the advantages of IRS and relay to serve the wireless
communication network is crucial, which has already attracted extensive
attention from both academia and industry [10, 11, 12, 13]. In [10], an IRS-
aided DF relay network was considered. To optimize the beamforming at relay
station (RS) and phase at IRS, three methods of maximizing receive power,
i.e., using alternately iterative structure, null-space projection based plus
maximum ratio combining (MRC) and IRS element selection based plus MRC, were
respectively proposed for rate improvement and coverage extension. For normal
communication between end users in [11], a two-way AF relay network assisted
by two IRSs was proposed, where alternatively optimizing phase at the two IRSs
to maximize sum rate. It demonstrated that the efficiency and sum rate
performance surpass that of only relay-aided system. [12] presented
cooperative relay networks with IRS, where reflection amplitudes changes with
the discrete phase. A deep reinforcement learning (DRL) scheme with low
complexity was proposed to optimize relay selection and IRS reflection
coefficient. It was shown that DRL outperformed random relay selection. In
[13], the authors investigated an IRS-aided two-way AF relay network, where
phase was solved by signal-to-noise ratio (SNR)-upper-bound-maximization or
genetic-SNR-maximization and beamforming was achieved by optimizing
beamforming or maximum-ratio beamforming. It demonstrated the proposed
algorithms evidently performed better than random phase.

However, all the above literature focused on the optimization of beamforming
at RS and phase at IRS, but did not consider power allocation (PA) between
users and RS. PA is an efficient way to improve the sum rate, which plays an
important role in sum rate. As far as we know, there is little research work
on the PA of IRS-aided two-way DF relay network. This motivates us to pay
attention to the PA under the total power constraint. Our main contributions
are summarized as follows:

  1. 1.

To improve the sum rate performance of an IRS-aided two-way DF relay network,
a PA strategy of maximizing sum rate (Max-SR) method is proposed to optimize
the PA factors of two users and RS. Since the constraints are non-convex, the
upper bound of the product of two PA factors and the first-order Taylor
approximation of quadratic of PA factor are derived, so that the constraints
can be converted into convex. The simulation results prove that the proposed
Max-SR method can harvest up to 13.8% sum rate gain over equal power
allocation (EPA) method.

  2. 2.

To achieve a higher sum rate, two high-performance schemes, namely maximizing
minimum sum rate (Max-Min-SR) and maximizing sum rate with rate constraint
(Max-SR-RC), are presented. In Max-Min-SR method, all the constraints are
convex with few converted operations. In Max-SR-RC method, the first-order
Taylor approximation is also applied to constraint transformation. Simulation
results show that the best method among the proposed three methods and EPA is
Max-SR-RC, which achieves at least 10% rate gain. Followed by Max-Min-SR
method, the sum rate gain harvested is up to 15.4%.

The remainder of this paper is organized as follows. In Section II, an IRS-
aided two-way DF relay network is described. In Section III, three high-
performance schemes for a better sum rate performance of the proposed network
is demonstrated. Simulation results are presented in Section IV, and
conclusions are drawn in Section V.

_Notation_ : Scalars, vectors and matrices are respectively represented by
letters of lower case, bold lower case, and bold upper case. $(\cdot)^{T}$ and
$(\cdot)^{H}$ stand for transpose and conjugate transpose, respectively.
$\mathbb{E}\\{\cdot\\}$ and $\|\cdot\|$ denote expectation operation and
2-norm, respectively. The sign $\textbf{I}_{M}$ is the $M\times M$ identity
matrix, and $\oplus$ represents exclusive or operation of two decoded symbols.

##  II System Model

Figure 1: System Model.

As shown in Fig. 1, we consider an IRS-aided two-way relay network. The
network comprises a half-duplex two-way DF RS with $M$ transmit antennas, an
IRS with $N$ reflecting elements, and two single-antenna users. Two users are
respectively denoted by $U_{1}$ and $U_{2}$, which mutually exchange
information with the help of IRS and RS. Due to path loss, the power of
signals reflected by the IRS twice or more are such weak that they can be
ignored. In the first time slot, the received signal at RS can be expressed as

$\displaystyle\textbf{y}_{r}=\sqrt{\beta_{1}{P}}(\textbf{h}_{1r}+\textbf{H}_{ir}{\bm{\Theta}_{1}}\textbf{h}_{1i})x_{1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{\beta_{2}{P}}(\textbf{h}_{2r}+\textbf{H}_{ir}{\bm{\Theta}_{1}}\textbf{h}_{2i})x_{2}+\textbf{n}_{r},$
(1)

where $x_{1}$ and $x_{2}$ are the independent transmit signal from $U_{1}$ and
$U_{2}$, respectively. $P$ is the total transmission power and limited,
$\beta_{1}$ and $\beta_{2}$ are the power allocation parameters of $U_{1}$ and
$U_{2}$. $\mathbb{E}\\{x_{1}^{H}{x_{1}}\\}=1$ and
$\mathbb{E}\\{x_{2}^{H}{x_{2}}\\}=1$. Without loss of generality, we assume a
Rayleigh fading environment [14]. Let $\textbf{h}_{1r}$$\in\mathbb{C}^{M\times
1}$, $\textbf{H}_{ir}\in\mathbb{C}^{M\times N}$ and
$\textbf{h}_{1i}\in\mathbb{C}^{N\times 1}$ represent the channels from $U_{1}$
to RS, from IRS to RS, and from $U_{1}$ to IRS,
$\textbf{h}_{2r}$$\in\mathbb{C}^{M\times 1}$ and
$\textbf{h}_{2i}\in\mathbb{C}^{N\times 1}$ represent the channels from $U_{2}$
to RS and from $U_{2}$ to IRS. $\bm{\Theta}_{1}$ is the diagonal reflection-
coefficient matrix of IRS, which is denoted as
$\bm{\Theta}_{1}=\text{diag}(e^{j\theta_{1}},\cdots,e^{j\theta_{N}})$,
$\theta_{i}\in(0,{2\pi}]$ is the phase shift of the $i$th element.
$\textbf{n}_{r}\in\mathbb{C}^{M\times 1}$ is the additive white Gaussian noise
(AWGN) with distribution $\textbf{n}_{r}\sim\mathcal{CN}(0,\sigma_{r}^{2}{\bf
I}_{M})$. Considering $x_{2}$ as unknown interference, RS first decodes
$x_{1}$ to $\widetilde{x}_{1}$. Then the contribution of $x_{1}$ can be
eliminated from (II), and $x_{2}$ can be decoded to $\widetilde{x}_{2}$.

In the second time slot, network coding is employed for $\widetilde{x}_{1}$
and $\widetilde{x}_{2}$, and combining them into a new signal, which is
represented as

${x}_{r}=\widetilde{x}_{1}\oplus\widetilde{x}_{2}.$ (2)

Each user decodes ${x}_{r}$ to $\widetilde{x}_{r}$, and rebuild the symbol
sent by the other user (i.e, $\overline{x}_{2}=\widetilde{x}_{r}\oplus{x}_{1}$
or $\overline{x}_{1}=\widetilde{x}_{r}\oplus{x}_{2}$). It is assumed that the
channel reciprocity holds, i.e., the channels in the second-time slot can be
represented as the conjugate transpose of the channels in the first-time slot.
After self-interference cancellation, the received signal at $U_{1}$ is
written by

$y_{1}=\sqrt{\beta_{3}{P}}(\textbf{h}_{1r}^{H}+\textbf{h}_{1i}^{H}{\bm{\Theta}_{2}}\textbf{H}_{ir}^{H})\overline{\textbf{x}}_{2}+n_{1},$
(3)

where
$\overline{\textbf{x}}_{2}=\overline{x}_{2}[1,1,\cdot\cdot\cdot,1]^{T}\in\mathbb{C}^{M\times
1}$. Similarly, the received signal at $U_{2}$ is given by

$y_{2}=\sqrt{\beta_{3}{P}}(\textbf{h}_{2r}^{H}+\textbf{h}_{2i}^{H}{\bm{\Theta}_{2}}\textbf{H}_{ir}^{H})\overline{\textbf{x}}_{1}+n_{2},$
(4)

where
$\overline{\textbf{x}}_{1}=\overline{x}_{1}[1,1,\cdot\cdot\cdot,1]^{T}\in\mathbb{C}^{M\times
1}$, $\beta_{3}$ is the power allocation parameter of RS, the diagonal
reflection-coefficient matrix of IRS is represented as
${\bm{\Theta}}_{2}=\text{diag}\left(e^{j{\theta_{21}}},\cdots,e^{j{\theta_{2N}}}\right)$,
${\theta_{2i}}\in\left(0\right.,\left.{2\pi}\right]$ is the phase shift of the
$i$th element. $n_{1}$ and $n_{2}$ are the AWGN with distribution
$n_{1}\sim\mathcal{CN}\left(0,{\sigma_{1}^{2}}\right)$ and
$n_{2}\sim\mathcal{CN}\left(0,{\sigma_{2}^{2}}\right)$, respectively. The
achievable rate of $U_{1}$-RS-$U_{2}$ link can be expressed as follows

$R_{12}=\text{min}\\{R_{1ir},R_{ri2}\\},$ (5)

where $R_{1ir}$ and $R_{ri2}$ are the rates of $U_{1}$-RS and RS-$U_{2}$ link,
respectively,

$R_{1ir}=\frac{1}{2}\log_{2}(1+\gamma_{1}\beta_{1}{P}),R_{ri2}=\frac{1}{2}\log_{2}(1+\gamma_{2}\beta_{3}{P}),$
(6)

and

$\gamma_{1}=\frac{\|\textbf{h}_{1r}+\textbf{H}_{ir}{\bm{\Theta}_{1}}\textbf{h}_{1i}\|^{2}}{\sigma_{r}^{2}},\gamma_{2}=\frac{\|\textbf{h}_{2r}^{H}+\textbf{h}_{2i}^{H}{\bm{\Theta}_{2}}\textbf{H}_{ir}^{H}\|^{2}}{\sigma_{2}^{2}}.$
(7)

Similarly, the achievable rate of $U_{2}$-RS-$U_{1}$ link can be represented
as follows

$R_{21}=\text{min}\\{R_{2ir},R_{ri1}\\},$ (8)

where $R_{2ir}$ and $R_{ri1}$ are the rates of $U_{2}$-RS and RS-$U_{1}$ link,
respectively,

$R_{2ir}=\frac{1}{2}\log_{2}(1+\gamma_{3}\beta_{2}{P}),R_{ri1}=\frac{1}{2}\log_{2}(1+\gamma_{4}\beta_{3}{P}),$
(9)

and

$\gamma_{3}=\frac{\|\textbf{h}_{2r}+\textbf{H}_{ir}{\bm{\Theta}_{1}}\textbf{h}_{2i}\|^{2}}{\sigma_{r}^{2}},\gamma_{4}=\frac{\|\textbf{h}_{1r}^{H}+\textbf{h}_{1i}^{H}{\bm{\Theta}_{2}}\textbf{H}_{ir}^{H}\|^{2}}{\sigma_{1}^{2}}.$
(10)

The achievable multiple access channel (MAC) rate of $U_{1}$-RS and $U_{2}$-RS
can be represented as follows

$R_{MAC}=\frac{1}{2}\log_{2}(1+\gamma_{1}\beta_{1}{P}+\gamma_{3}\beta_{2}{P}).$
(11)

Therefore, the achievable sum rate of the proposed system is defined as
follows

$\displaystyle R=\text{min}\\{R_{12}+R_{21},R_{MAC}\\}$
$\displaystyle=\text{min}\\{\text{min}\\{R_{1ir},R_{ri2}\\}+\text{min}\\{R_{2ir},R_{ri1}\\},R_{MAC}\\}.$
(12)

##  III Proposed three PA methods

In this section, we focus on the investigation of PA methods to maximize $R$,
and the optimization problem is casted as

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3},{\bm{\Theta}_{1}},{\bm{\Theta}_{2}}}~{}~{}~{}R$
(13a)
$\displaystyle~{}~{}~{}~{}~{}~{}\text{s.t.}~{}~{}0<\beta_{1},\beta_{2},\beta_{3}<1,~{}~{}\beta_{1}+\beta_{2}+\beta_{3}=1,$
(13b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|\bm{\Theta}_{1}(i,i)|=1,|\bm{\Theta}_{2}(i,i)|=1,\forall
i=1,\cdots,N,$ (13c)

where $\bm{\Theta}_{1}$ and $\bm{\Theta}_{2}$ are obtained by maximizing the
sum rate via general power iterative in [15]. In this paper, $\bm{\Theta}_{1}$
and $\bm{\Theta}_{2}$ refer directly to [15]. The above optimization problem
is simplified to

$\max\limits_{\beta_{1},\beta_{2},\beta_{3}}~{}~{}R~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{s.t.}~{}~{}~{}~{}(\text{\ref{R_1}}).$
(14)

To solve the optimization problem, three PA schemes are proposed, which are
Max-SR, Max-Min-SR and Max-SR-RC, respectively, and the related details are as
follow.

###  III-A Proposed Max-SR

The system sum rate $R$ is expanded as follows

$\displaystyle
R=\text{min}\\{\text{min}\\{R_{1ir}+R_{2ir},R_{1ir}+R_{ri1},R_{ri2}+R_{2ir}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},R_{ri2}+R_{ri1}\\},R_{MAC}\\}$
$\displaystyle~{}~{}=\text{min}\\{R_{1ir}+R_{2ir},R_{1ir}+R_{ri1},R_{ri2}+R_{2ir}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},R_{ri2}+R_{ri1},R_{MAC}\\}.$
(15)

Clearly, $R_{1ir}+R_{2ir}>R_{MAC}$, so the case of $R_{1ir}+R_{2ir}$ is
excluded directly. The above equation is reduced to

$R=\text{min}\\{R_{1ir}+R_{ri1},R_{ri2}+R_{2ir},R_{ri2}+R_{ri1},R_{MAC}\\},$
(16)

the optimization problem is further recasted as

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3},R}~{}~{}R$ (17a)
$\displaystyle~{}~{}~{}~{}\text{s.t.}~{}~{}~{}~{}~{}~{}~{}0<\beta_{1},\beta_{2},\beta_{3}<1,~{}~{}\beta_{1}+\beta_{2}+\beta_{3}=1,$
(17b) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R\leq
R_{1ir}+R_{ri1},$ (17c)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R\leq
R_{ri2}+R_{2ir},$ (17d)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R\leq
R_{ri2}+R_{ri1},$ (17e)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R\leq R_{MAC},$ (17f)

where

$\displaystyle
R_{1ir}+R_{ri1}=\frac{1}{2}\log_{2}(1+\gamma_{1}\beta_{1}{P}+\gamma_{4}\beta_{3}{P}+\gamma_{1}\gamma_{4}\beta_{1}\beta_{3}{P^{2}}),$
(18a) $\displaystyle
R_{ri2}+R_{2ir}=\frac{1}{2}\log_{2}(1+\gamma_{3}\beta_{2}{P}+\gamma_{2}\beta_{3}{P}+\gamma_{2}\gamma_{3}\beta_{2}\beta_{3}{P^{2}}),$
(18b) $\displaystyle
R_{ri2}+R_{ri1}=\frac{1}{2}\log_{2}[1+\beta_{3}{P}(\gamma_{2}+\gamma_{4})+\gamma_{2}\gamma_{4}{\beta_{3}}^{2}{P^{2}}].$
(18c)

The above optimization is non-convex due to the non-convex constraints (17c),
(17d) and (17e), it is necessary to convert (17c), (17d) and (17e) to convex.
Inserting $\beta_{3}=1-\beta_{1}-\beta_{2}$ back into $R_{1ir}+R_{ri1}$ yields
the following inequality

$\displaystyle 2^{2R}\leq
1+\gamma_{4}P+\beta_{1}(\gamma_{1}P-\gamma_{4}P+\gamma_{1}\gamma_{4}P^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\gamma_{1}\gamma_{4}\beta_{1}^{2}P^{2}-\gamma_{4}\beta_{2}P-\gamma_{1}\gamma_{4}\beta_{1}\beta_{2}P^{2}.$
(19)

Since $\beta_{1}\beta_{2}\leq\frac{1}{2}(\beta_{1}^{2}+\beta_{2}^{2})$, the
lower-bound of $-\beta_{1}\beta_{2}$ is
$-\frac{1}{2}(\beta_{1}^{2}+\beta_{2}^{2})$, which further yields

$\displaystyle 2^{2R}\leq
1+\gamma_{4}P+\beta_{1}(\gamma_{1}P-\gamma_{4}P+\gamma_{1}\gamma_{4}P^{2})-\gamma_{4}\beta_{2}P$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{\gamma_{1}\gamma_{4}(3\beta_{1}^{2}+\beta_{2}^{2})P^{2}}{2},$
(20)

which is a convex constraint. In the same manner, (17d) can be rewritten as

$\displaystyle 2^{2R}\leq
1+\gamma_{2}P+\beta_{2}(\gamma_{3}P-\gamma_{2}P+\gamma_{2}\gamma_{3}P^{2})-\gamma_{2}\beta_{1}P$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{\gamma_{2}\gamma_{3}(\beta_{1}^{2}+3\beta_{2}^{2})P^{2}}{2}.$
(21)

Similarly, (17e) can be written in the following form

$2^{2R}\leq
1+\beta_{3}P(\gamma_{2}+\gamma_{4})+\gamma_{2}\gamma_{4}\beta_{3}^{2}{P^{2}},$
(22)

which is still a non-convex constraint.
$\gamma_{2}\gamma_{4}\beta_{3}^{2}{P^{2}}$ is a convex function, its low bound
can be expressed by the first-order Taylor expansion. The first-order Taylor
expansion of $\gamma_{2}\gamma_{4}\beta_{3}^{2}{P^{2}}$ at feasible point
$\beta_{3t}$ is given by

$\gamma_{2}\gamma_{4}\beta_{3}^{2}{P^{2}}\geq\gamma_{2}\gamma_{4}\beta_{3t}^{2}{P^{2}}+2\gamma_{2}\gamma_{4}\beta_{3t}{P^{2}}(\beta_{3}-\beta_{3t}).$
(23)

Combining (22) and (23), (17e) further can be converted into

$2^{2R}\leq
1+\gamma_{2}\gamma_{4}\beta_{3t}^{2}{P^{2}}+2\gamma_{2}\gamma_{4}\beta_{3t}{P^{2}}(\beta_{3}-\beta_{3t})+\beta_{3}P(\gamma_{2}+\gamma_{4}).$
(24)

Therefore, the optimization problem is further reformulated as

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3},R}~{}~{}R$ (25a)
$\displaystyle~{}~{}~{}~{}\text{s.t.}~{}~{}~{}~{}~{}~{}~{}0<\beta_{1},\beta_{2},\beta_{3}<1,~{}~{}\beta_{1}+\beta_{2}+\beta_{3}=1,$
(25b) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R\leq
R_{MAC},~{}~{}(\ref{18d}),~{}~{}(\ref{18e}),~{}~{}(\ref{18f}),$ (25c)

which is a convex optimization problem and can be solved efficiently via CVX.

###  III-B Proposed Max-Min-SR

In the subsection III-A, Max-SR is proposed to enhance the rate performance.
Meanwhile, Max-SR needs operations to convert constraints from non-convex to
convex. To further improve the sum rate gain, Max-Min-SR method is proposed,
where two intermediate variables $R_{1}$ and $R_{2}$ are introduced. The
optimization problem can be given by

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3}}R=\text{min}\\{\text{min}\\{R_{1ir},R_{ri2}\\}+\text{min}\\{R_{2ir},R_{ri1}\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},R_{MAC}\\}$
(26a)
$\displaystyle~{}~{}~{}\text{s.t.}~{}~{}~{}0<\beta_{1},\beta_{2},\beta_{3}<1,~{}~{}\beta_{1}+\beta_{2}+\beta_{3}=1.$
(26b)

It is defined that $R_{1ir}\geq R_{1}$, $R_{ri2}\geq R_{1}$, $R_{2ir}\geq
R_{2}$, $R_{ri1}\geq R_{2}$, $R_{MAC}\geq R$ and $R_{1}+R_{2}\geq R$. The
above optimization problem is reformulated as

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3},R_{1},R_{2},R}R$
(27a)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\text{s.t.}~{}0<\beta_{1},\beta_{2},\beta_{3}<1,~{}~{}\beta_{1}+\beta_{2}+\beta_{3}=1,$
(27b) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2{R_{1}}}\leq
1+\gamma_{1}\beta_{1}P,~{}~{}2^{2{R_{1}}}\leq 1+\gamma_{2}\beta_{3}P,$ (27c)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2{R_{2}}}\leq
1+\gamma_{3}\beta_{2}P,~{}~{}2^{2{R_{2}}}\leq 1+\gamma_{4}\beta_{3}P,$ (27d)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2R}\leq
1+\gamma_{1}\beta_{1}P+\gamma_{3}\beta_{2}P,~{}R\leq R_{1}+R_{2},$ (27e)

where the object function and each constraint are convex. Thus it is a convex
optimization problem, which can also be solved by CVX directly.

###  III-C Proposed Max-SR-RC

In fact, there exist asymmetry of two-way channel quality and two users’
demand in the IRS-aided two-way DF relay network, thereby the asymmetry
between $R_{12}$ and $R_{21}$ is generated. In this subsection, we make an
investigation of PA in the case of $R_{12}=\mu R_{21}$, which is called Max-
SR-RC method. The optimization problem is given by

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3}}R$ (28a)
$\displaystyle~{}~{}\text{s.t.}~{}~{}~{}~{}~{}0<\beta_{1},\beta_{2},\beta_{3}<1,~{}~{}\beta_{1}+\beta_{2}+\beta_{3}=1,$
(28b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0<\mu,~{}~{}R_{12}={\mu}R_{21},$
(28c)

where $\mu$ is a constant. Substituting (28c) into the object function and
expanding it, the above problem can be rewritten as

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3}}\text{min}\\{\frac{1}{2}\log_{2}(1+\gamma_{3}\beta_{2}P)^{1+\mu},\frac{1}{2}\log_{2}(1+\gamma_{4}\beta_{3}P)^{1+\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},\frac{1}{2}\log_{2}(1+\gamma_{1}\beta_{1}P+\gamma_{3}\beta_{2}P)\\}$
(29a)
$\displaystyle~{}~{}~{}\text{s.t.}~{}~{}~{}~{}0<\mu,~{}~{}(\text{\ref{C_1}}).$
(29b)

Since it is similar to Max-SR method, the above problem can be further
transformed as

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3},R}~{}~{}R$ (30a)
$\displaystyle~{}~{}~{}~{}\text{s.t.}~{}~{}~{}~{}~{}~{}~{}2^{2R}\leq(1+\gamma_{3}\beta_{2}P)^{1+\mu},$
(30b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2R}\leq(1+\gamma_{4}\beta_{3}P)^{1+\mu},$
(30c) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2R}\leq
1+\gamma_{1}\beta_{1}P+\gamma_{3}\beta_{2}P,~{}~{}(\text{\ref{D_1}}),$ (30d)

where (30b) and (30c) are non-convex constraints. In the same manner, the low
bounds of convex functions $(1+\gamma_{3}\beta_{2}P)^{1+\mu}$ and
$(1+\gamma_{4}\beta_{3}P)^{1+\mu}$ can be achieved through the first-order
Taylor expansion. The details are as follow

$\displaystyle(1+\gamma_{3}\beta_{2}P)^{1+\mu}\geq(1+\gamma_{3}{\beta_{2t}}P)^{1+\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(1+\mu)(1+\gamma_{3}{\beta_{2t}}P)^{\mu}(\beta_{2}-{\beta_{2t}}),$
(31)
$\displaystyle(1+\gamma_{4}\beta_{3}P)^{1+\mu}\geq(1+\gamma_{4}{\beta_{3t}}P)^{1+\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(1+\mu)(1+\gamma_{4}{\beta_{3t}}P)^{\mu}(\beta_{3}-{\beta_{3t}}),$
(32)

where $\beta_{2t}$ and $\beta_{3t}$ are feasible points. Substituting the two
low bounds into the above optimization problem, respectively, yields

$\displaystyle\max\limits_{\beta_{1},\beta_{2},\beta_{3},R}R$ (33a)
$\displaystyle~{}~{}~{}~{}\text{s.t.}~{}~{}~{}~{}~{}2^{2R}\leq(1+\gamma_{3}{\beta_{2t}}P)^{1+\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(1+\mu)(1+\gamma_{3}{\beta_{2t}}P)^{\mu}(\beta_{2}-{\beta_{2t}}),$
(33b)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2R}\leq(1+\gamma_{4}{\beta_{3t}}P)^{1+\mu}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(1+\mu)(1+\gamma_{4}{\beta_{3t}}P)^{\mu}(\beta_{3}-{\beta_{3t}}),$
(33c) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2^{2R}\leq
1+\gamma_{1}\beta_{1}P+\gamma_{3}\beta_{2}P,~{}~{}(\text{\ref{D_1}}),$ (33d)

which is also a convex optimization problem. Similarly, PA factors
$\beta_{1}$, $\beta_{2}$, $\beta_{3}$ and sum rate $R$ can be obtained.

##  IV Simulation And Numerical Results

In this section, numerical simulations are performed to evaluate and compare
the sum rate performance between the proposed three methods and EPA method.
Moreover, it is assumed that $U_{1}$, $U_{2}$, IRS and RS are located in
three-dimensional (3D) space, the positions of $U_{1}$, $U_{2}$, IRS and RS
are given as (0, 0, 0), (0, 100m, 0), ($-$10m, 50m, 20m) and (10m, 50m, 10m),
respectively. We take a more realistic environment into account, and assume
that all channels follow large-scale fading, which contains shadow fading. The
path loss model is
$PL(d)=PL_{0}-10{\alpha}\text{log}_{10}(\frac{d}{d_{0}})-X_{\sigma}$, where
$PL_{0}=-20\text{log}_{10}(\frac{4{\pi}d_{0}}{\lambda})$ is the path loss at
the reference distance $d_{0}$, $\lambda$ is wavelength, $\alpha$ is the path
loss exponent, and $X_{\sigma}$ is a Gaussian random shadow variable with
distribution $X_{\sigma}\sim\mathcal{CN}(0,\sigma^{2})$. The path loss
exponents associated with IRS are 2.1, those of $U_{1}$-RS and $U_{2}$-RS
links are 2.3. The remaining system parameters are set as follows: $f_{c}=$
1.5GHz, $\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{r}^{2}=-$80dBm.

Figure 2: Achievable sum rate of proposed Max-SR-RC versus $\mu$ with $M=$ 4,
$N=$ 16 and $\sigma=3$dB.
Figure 3: Achievable sum rate versus total power with $M=$ 4, $N=$ 16 and
$\sigma=3$dB.
Figure 4: Achievable sum rate versus $\sigma$ with total power $=$ 40dBm, $M=$
4 and $N=$ 16.

Fig. 2 illustrates the achievable sum rate of Max-SR-RC method versus $\mu$
with different total power: $P=\\{$0dBm, 10dBm, 20dBm, 30dBm$\\}$. It can be
seen that as $\mu$ increases, the sum rate performance gradually increases.
Until $\mu\geq$ 3, the achievable sum rate tends to be stable. For
convenience, $\mu$ is defined as 3.

Fig. 3 depicts the histogram of achievable sum rate versus total power with
$M=$ 4, $N=$ 16 and $\sigma=$ 3dB. It can be seen that the proposed three
methods perform better than EPA method. For instance, when total power equals
10dBm, the proposed worst method, Max-SR, can harvest up to 5.3% rate gain
over EPA method. The best method, Max-SR-RC, approximately has a 37.9% rate
gain over EPA method.

Fig. 4 shows the corresponding histogram of achievable sum rate versus
$\sigma$ with total power $=$ 40dBm, $M=$ 4 and $N=$ 16\. When $\sigma=$ 5dB,
it can be observed that the achievable sum rate performance improvements over
EPA method are 13.8%, 15.4% and 45.2%, respectively. Moreover, it indicates
that as $\sigma$ increases, the sum rate performance gain becomes more
obvious, which is very attractive. However, greater shadowing fading is formed
because of the increase of $\sigma$, which sharply deteriorates the sum rate
performance.

##  V Conclusions

In this paper, in order to improve the achievable sum rate of an IRS-aided
two-way DF relay wireless network, three high-performance PA schemes, namely
Max-SR, Max-Min-SR and Max-SR-RC, were proposed. Simulation results showed
that the proposed three methods can harvest better rate gain over EPA method
in terms of sum rate performance. Because of excellent rate performance, the
proposed Max-SR-RC method is very attractive. Additionally, the rate increases
with total power, while decreases with $\sigma$.

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# Crystal and electronic structure of a quasi-two-dimensional semiconductor
Mg3Si2Te6

Chaoxin Huang  The authors contributed equally to this work. Center for
Neutron Science and Technology, Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen
University, Guangzhou, 510275, China  Benyuan Cheng  The authors contributed
equally to this work. Shanghai Institute of Laser Plasma, Shanghai 201800,
China  Center for High Pressure Science and Technology Advanced Research,
Shanghai 201203, China  Yunwei Zhang  Center for Neutron Science and
Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and
Devices, School of Physics, Sun Yat-Sen University, Guangzhou, 510275, China
Long Jiang  Instrumentation Analysis and Research Center, Sun Yat-Sen
UniVersity, Guangzhou 510275, China  Lisi Li  Mengwu Huo  Hui Liu  Xing Huang
Feixiang Liang  Lan Chen  Hualei Sun  Meng Wang<EMAIL_ADDRESS>Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory
of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen
University, Guangzhou, 510275, China

###### Abstract

We report the synthesis and characterization of a Si-based ternary
semiconductor Mg3Si2Te6, which exhibits a quasi-two-dimensional structure,
where the trigonal Mg2Si2Te6 layers are separated by Mg ions. Ultraviolet-
visible absorption spectroscopy and density functional theory calculations
were performed to investigate the electronic structure. The experimentally
determined direct band gap is 1.39 eV, consistent with the value of the
density function theory calculations. Our results reveal that Mg3Si2Te6 is a
direct gap semiconductor with a relatively narrow gap, which is a potential
candidate for infrared optoelectronic devices.

##  I INTRODUCTION

Narrow-gap semiconductors (NGS) play an important role in optoelectronic and
magnetic devices. Because of their unique advantages, the properties and
applications of NGS have been widely studied for decades in a variety of
fields, such as magnetic field sensors, photovoltaics, infrared
photodetectors, lasers and so onElliott (1998); Maier and Hesse (1980);
Stradling (1996); Xie  _et al._ (2021); Zhang  _et al._ (2017). Magnetic field
sensors, including magnetoresistors and Hall sensors, are generally used in
magnetic recording and magnetic measurement technologiesRipka and Janosek
(2010); Heremans  _et al._ (1993). Materials for magnetic field sensors are
doped semiconductors with high electronic density of states and mobility, such
as InSb and InAs which are extremely sensitive to magnetic fieldSolin  _et
al._ (2000); Berus  _et al._ (2004); Mihajlović  _et al._ (2005). Conventional
photovoltaic materials mostly use wide-gap semiconductors that utilize photons
in the ultraviolet and visible regions, limiting the conversion efficiency.
Tuning the band gap is considered to be one of the effective solutionsCheng
and Yang (2020). NGS-based photoelectrodes can absorb light at longer
wavelengths and enhance the solar energy conversion efficiencyZheng  _et al._
(2019). As a more widely used field, infrared photodetectors, such as the
quantum dots-in-a-well photodetectors, have developed rapidly in recent
yearsDowns and Vandervelde (2013); Martyniuk and Rogalski (2008); Chen  _et
al._ (2018). In addition, NGS lasers have attracted more and more attention
due to their stability, tunability, simple structure, and many other
advantagesTournie and Baranov (2012); Harman and Melngailis (1974).

One of the most important parameters for photoelectric materials is the value
of the electronic band gap. Hg1-xCdxTe and Pb1-xSnxTe are among the hottest
researched NGS systems in recent years, whose gaps can be tuned by
compositions in a wide range, even close to zeroHarman and Melngailis (1974);
Rogalski (2005); Lei  _et al._ (2015); Piotrowski and Rogalski (2004).
Alloying is also an effective method to modulate the band gap, which is
beneficial for the application of the mid- and long-infrared
photodetectorsYang  _et al._ (2022). Although InSb and HgCdTe based detectors
have been well available in commercial applications, their poor flexibility,
instability of the material interface, complex manufacturing process, and low
temperature working environment limit their further developmentPiotrowski and
Rogalski (1998); Jiao  _et al._ (2022); Yang  _et al._ (2022). Therefore, it
is crucial to search for more NGS that are suitable for manufacturing stable
and efficient infrared detectors. Meanwhile, new NGS with excellent
performance also have potential applications in laser, photovoltaics, and
other magnetic devices.

In this work we report the successful synthesis of a Si-based nonmagnetic
semiconductor Mg3Si2Te6, whose structure is found to exhibit a quasi-two-
dimensional (quasi-2D) layered configuration. Fittings using a direct and
indirect band gap models to the ultraviolet-visible (UV-vis) absorption
spectra result in a direct gap of 1.39 eV or an indirect gap of 0.6 eV,
respectively. Electronic structure calculations using density functional
theory (DFT) yield a direct band gap of 1.2 eV, close to the direct gap fitted
from the UV-vis absorption spectra, revealing that Mg3Si2Te6 is a direct gap
semiconductor. The band gap is narrower than the energies of visible photons,
allowing Mg3Si2Te6 a potential candidate for infrared photoelectric material.

Figure 1: Crystal structure of Mg3Si2Te6 (space group: $P\overline{3}1c$) viewed (a) perpendicular to the $c$-axis and (b) parallel to the $c$-axis, respectively. The orange, blue, and grey balls represent Mg, Si, and Te.  Table 1: Parameters of Mg3Si2Te6 refined from single-crystal XRD at 253K. Empirical formula | Mg3Si2Te6
---|---
Formula weight | 894.69
Temperature | 252.99(10) K
Crystal system | trigonal
Space group | $P\overline{3}1c$
Unit-cell parameters |  $a$ = $b$ = 7.0642(9) Å
|  $c$ = 14.464(2) Å
|  $\alpha$ = $\beta$ = 90∘
|  $\gamma$ = 120∘
Atomic parameters |
Mg1 | 4f (1/3, 2/3, 0.4987(5))
Mg2 | 2d (2/3, 4/3, 1/4)
Si | 4e (0, 1, 0.4192(4))
Te | 12i ($x$, $y$, $z$)
|  $x$ = 0.3402(1), $y$ = 0.9990(2),
|  $z$ = 0.3733(1)
Volume | 625.11(18) Å3
Density | 4.753 g/cm3
Absorp. coeff. | 14.102 mm-1
$F$(000) | 725.0
Crystal size | 0.09 $\times$ 0.04 $\times$ 0.02 mm3
Radiation | Mo Kα ($\lambda$ = 0.7107 Å )
2$\Theta$ range for data collection | 5.632∘ to 53.96∘
Index ranges | -9 $\leq$ h $\leq$ 7, -9 $\leq$ k $\leq$ 9,
| -18 $\leq$ l $\leq$ 18
Reflections collected | 3733
Independent reflections | 439
Data/restraints/parameters | 439/0/18
Goodness-of-fit on $F$2 | 0.904
Final R indexes [$I$ $\geq$ 2$\sigma$(I)] |  $R$1 = 0.0391, $wR$2 = 0.0790
Final R indexes [all data] |  $R$1 = 0.0767, $wR$2 = 0.1014
Largest diff. peak/hole/_e_ Å-3 | 1.38/ - 0.96

##  II EXPERIMENT AND CALCULATION

Single crystal samples of Mg3Si2Te6 were grown by a self-flux methodYin  _et
al._ (2020); Sun  _et al._ (2021); Li  _et al._ (2021). Mg (99.99$\%$) robs,
Si (99.99$\%$) powders, and Te blocks (99.99$\%$) were mixed in a molar ratio
of $3:2:6$ after rough grinding, then sealed in an evacuated quartz ampoule.
The ampoule was firstly heated to 600 ∘C in 10 h and held for 15 h, then
heated to 1050 ∘C in 20 h and dwelled for 10 h, followed by a slow cooling to
750 ∘C in 150 h before the furnace was shut down. Shiny plate-like single
crystals were obtained. The samples are air sensitive and were stored in an
argon-filled glove box to minimize the exposure to air.

Figure 2:  (a) A zoomed-in view on the surface of the Mg3Si2Te6 single crystal
taken from SEM. The three circles mark the measured spots. (b) An EDS spectrum
of the spot 1 in (a). The elements represented by different characteristic
peaks are indicated. Small amounts of carbon and oxygen may derive from the
background of conductive adhesive tape. The inset is a photo of the single
crystal under an optical microscope.

Single-crystal x-ray diffraction (XRD) measurements on a single-crystal x-ray
diffractometer (SuperNova, Rigaku), a scanning electron microscopy (SEM)
photography, and an energy dispersive x-ray spectroscopy (EDS) (EVO, Zeiss)
were employed to determine the crystal structure, morphology and composition.
Resistivity was measured using the standard four-probe method on single
crystals with a typical size of 3 $\times 1\times 0.5$ mm3 on a physical
property measurement system (PPMS) (Quantum Design). UV-vis absorption
spectroscopy measurements were conducted using an Ocean Optics DH-2000-BAL
spectrometer. The sample was pre-peeled into thin flakes with right thickness
and then loaded into a diamond anvil cell (DAC) to avoid oxidation. The
outboard ring of the DAC is the T301 steel gasket, which was pre-indented and
drilled a hole with a diameter of 150 $\mu$m to serve as the sample chamber.
Each spectrum was collected for 2 seconds with wavelengthes ranged from 180 to
860 nm. The signal from background was obtained by shining the beam on a spot
inside the sample chamber but away from the sample.

Electronic band structure was calculated using the DFT within the Perdew-
Burke-Ernzerhof (PBE) exchange-correlation as implemented in the Vienna
$Ab~{}initio$ Simulation Package (VASP) codePerdew  _et al._ (1996); Vargas
Hernández (2020). A plane wave energy cutoff of 600 eV and dense meshes of
Monkhorst-Pack $k$-points were used to ensure that all calculations are well
converged to 1 meV/atom.

Figure 3: (a) Resistivity of Mg3Si2Te6 in the temperature range of
328$\sim$400 K. (b) $\ln\rho$ as a function of $1000/T$. The red line is a
fitting using the thermal activation-energy modol $\rho$($T$) =
$\rho$0exp($E$a/$k$B$T$).

##  III RESULTS AND DISCUSSION

The structure of Mg3Si2Te6 is refined from XRD measurements on single
crystals. The structural parameters are summarized in Table 1. Mg3Si2Te6
crystalizes in the trigonal space group $P\overline{3}1c$ with $a$ = 7.0642(9)
Å and $c$ = 14.464(2) Å at 253 K and exhibits a quasi-2D structure. The
crystal structures viewed from different directions are illustrated in Fig. 1.
The layer of the $ab$ plane constituting the formula as Mg2Si2Te6 consists of
edge shared MgTe6 octahedra and Si-Si dimers. The Mg2Si2Te6 layer is
isomorphic to a widely studied 2D van der Waals ferromagnetic compound
Cr2Si2Te6Carteaux  _et al._ (1995); Cai  _et al._ (2020). The layers are
linked by Mg2 that is half of Mg1, constituting Mg3Si2Te6. Mg3Si2Te6 is
isostructural to Mn3Si2Te6Vincent  _et al._ (1986) and Mn3Si2Se6May  _et al._
(2020) which are quasi-2D semiconductors with enriched properties such as
ferrimagnetism and large colossal magnetoresistanceNi  _et al._ (2021); Seo
_et al._ (2021).

To determine the composition of the samples, we performed EDS measurements on
Mg3Si2Te6 single crystals. Figure 2(a) is a zoomed-in view of the measured
crystal in the $ab$ plane taken from SEM. It is obvious that the crystal is
layered with some Te flux on the surface. By normalizing the content of Mg to
be 3, the composition determined from EDS is Mg3Si1.87(4)Te6.8(2). The ratio
of Mg to Si ratio is close to expectation. The exceeding content of Te of 13%
can be attributed to the excess Te flux on the surface.

Figure 4: (a) UV-vis absorption spectra of Mg3Si2Te6 in a DAC at room
temperature. (b) The Tauc plots of [$F$($R$∞)$h$$\nu$]1/n = $A$($h$$\nu$ \-
$E_{g}$) versus $h\nu$ with $n$ = 1/2 for direct band gap and (c) $n$ = 2 for
indirect band gap. The dotted red lines are fittings of the linear regions.

Figure 3 shows the resistivity measured between 328 and 400 K. The resistivity
below 328 K exceeding 106 $\Omega$cm is beyond the measured range of the
instrument. As shown in Fig. 3 (a), the resistivity exhibits an insulating
behavior, and no obvious anomaly can be observed up to 400 K. Figure 3 (b) is
a plot of the resistivity in $\ln\rho$ as a function of 1000/$T$. A fitting
using the thermal activation-energy model $\rho$($T$) = $\rho$0
exp($E$a/$k$B$T$) to the linear segment that corresponds to 350 to 400 K
results in $E_{a}=0.378$ eV. In the model, $\rho$0 is a prefactor, $k$B is the
Boltzmann constant, and $E_{a}$ is the thermal activation energy. The obtained
activation energy of 0.378 eV indicates that the energy gap of Mg3Si2Te6 may
be narrower than that of visible light of $1.64\sim 3.19$ eV. We note the
fitted thermal activation energy at $350\sim 400$ K may be much smaller than
the band gap.

Figure 5: Electronic structure calculation using DFT. The direct gap of
Mg3Si2Te6 is about 1.2 eV. PDOS is obtained by integrating density states of
the energy bands, and the electrons close to the fermi surface are mainly
provided by Te.

UV-vis absorption spectra were employed to study the electronic band gap of
Mg3Si2Te6. Five pieces of single crystals were measured separately. The
absorption spectra are shown in Fig. 4 (a), yielding a typical semiconducting
state with a strong absorption in the range of $700\sim 860$ nm, while the
rest of the short wavelength data are removed because of relatively high
noise. The band gap can be obtained by fitting the absorption spectra using
the Tauc relation that is [$F$($R$∞)$h$$\nu$]1/n = $A$($h$$\nu$ \- $E_{g}$),
where $F$($R$∞) is the Kubelka-Mubelka-Munk function, $h$ is the Planck
constant, $\nu$ is the frequency, $A$ is a prefactor, and $E_{g}$ is the band
gap energySarkar  _et al._ (2017). For a direct band gap semiconductor, $n$ is
1/2; for an indirect band gap semiconductor, $n$ is 2. We plot the Tauc
relations as a direct gap and indirect gap semiconductor in Figs. 4 (b) and 4
(c), respectively. The size of the band gap is extracted from the linear
regression at the inflection point and the obtained $h\nu$-intercept value is
taken as the band gap valueTauc  _et al._ (1966). The measured absorption
spectra could not distinguish the two models due to poor statistic. We fitted
the linear parts of five sets of data by adopting the Tauc relation and
$n=1/2$, resulting in the direct band gaps of 1.41, 1.43, 1.43, 1.37, and 1.31
eV. Thus, the direct gap is estimated to be 1.39(5) eV by averaging the fitted
values. Following the same procedure and adopting $n=2$, an indirect band gap
of 0.6(2) eV is obtained. The difference in the calculated energy gaps between
different data sets can be attributed to the inconsistency of thicknesses of
the samples, which causes calculation errors by affecting the transmittance
and reflectivity of the samples to ultraviolet light.

In order to distinguish the direct gap and indirect gap from the fittings of
the UV-vis absorption spectra, DFT calculations were performed to investigate
the electronic band structure. Figure 5 displays the band structures along
high symmetry directions in a Brillouin zone and partial density of states
(PDOS) of Mg3Si2Te6. The calculated data reveal a direct gap of $\sim$1.2 eV
at the $\Gamma$ point, close to the fitting results from the UV-vis absorption
spectra using the direct gap model. Thus, Mg3Si2Te6 can be considered as a
direct gap semiconductor with a gap size of $1.2\sim 1.39$ eV. This value is
larger than the thermal activation energy of 0.378 eV fitted from the
resistivity between 350 and 400 K. The outcomes may attribute to the fact that
the activation energy is fitted from resistance change at high temperature,
where thermal fluctuations increase the electron hopping capability, and the
gap seems to be narrowed. The direct gap size of Mg3Si2Te6 is smaller than the
energies of visible light. Through doping, temperature, and pressure, the band
gap of Mg3Si2Te6 may be further decreased, allowing Mg3Si2Te6 one of the
potential candidates for infrared photoelectric materials.

##  IV SUMMARY

In conclusion, we synthesized single crystals of Mg3Si2Te6 by self-flux
method, and characterized the structure, composition, resistivity, and
electronic band gap. Mg3Si2Te6 exhibits a quasi-2D structure constituted by
Mg2Si2Te6 layers (Mg1) that are linked by Mg2 atoms. The single crystals can
be mechanically cleaved and are sensitive to air. By combining the UV-vis
absorption spectra and DFT calculations, we demonstrate that Mg3Si2Te6 is a
direct band gap semiconductor with a gap size of $1.2\sim 1.39$ eV. This gap
size is close to that of silicon, and may allow Mg3Si2Te6 become one of the
candidates for infrared photoelectric or other functional materials by tuning
the energy level by rare earth ions doping. Moreover, the direct gap
characteristics may provide higher photon absorption efficiency compared with
traditional indirect-gap Si-based semiconductors. The absorption capacity and
conversion efficiency require further investigations.

##  V ACKNOWLEDGMENTS

Work was supported by the National Natural Science Foundation of China (Grants
No. 12174454, No. 11904414, No. 11904416, and No. 12104427), the Guangdong
Basic and Applied Basic Research Foundation (Grant No. 2021B1515120015), and
National Key Research and Development Program of China (Grant No.
2019YFA0705702).

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