GAN

    Generative Adversarial Network
DCGAN

    Deep Convolutional Generative Adversarial Network
SNGAN

    Spectral Normalization Generative Adversarial Network
SA-GAN

    Self-Attention Generative Adversarial Network
BigGAN

    Big Generative Adversarial Network
WGAN

    Wasserstein Generative Adversarial Network
WGAN-GP

    Wasserstein Generative Adversarial Network with Gradient Penalty
DGMR

    Deep Generative Model of Rainfall
VAE

    Variational Autoencoder
ASL

    Arterial Spin Labeling
MRI

    Magnetic Resonance Imaging
PET

    Positron Emission Tomography
ADNI

    Alzheimer’s Disease Neuroimaging Initiative
MSL

    Multiple Scale and Location
SVC

    Singular Value Clipping
LIDC

    The Lung Image Database Consortium image collection
TCIA

    The Cancer Imaging Archive
BRATS

    The Multimodal Brain Tumor Image Segmentation Benchmark
CT

    Computerized Tomography
HU

    Hounsfield Units
DICOM

    Digital Imaging and Communications in Medicine
FID

    Fréchet Inception Distance
EM

    Earth Mover
TWR

    Tournament Win Rate
MS-SSIM

    Multiple Scale Structural Similarity Index
bMMD2

    batch-wise Maximum Mean Discrepancy2
CAD

    Computer-Aided Diagnostics
TTUR

    Two Time-scale Update Rule
11institutetext: Department of Computer Science, University of Copenhagen,
Denmark
11email<EMAIL_ADDRESS>22institutetext: Department of Oncology,
Rigshospitalet, Denmark33institutetext: Department of Neuroscience, University
of Copenhagen, Denmark44institutetext: Faculty of Health and Medical Sciences,
University of Copenhagen, Copenhagen, Denmark

# Explicit Temporal Embedding in Deep Generative Latent Models for
Longitudinal Medical Image Synthesis

Julian Schön 1122 Raghavendra Selvan 1133 Lotte Nygård 22 Ivan Richter
Vogelius 2244 Jens Petersen 1122

###### Abstract

Medical imaging plays a vital role in modern diagnostics and treatment. The
temporal nature of disease or treatment progression often results in
longitudinal data. Due to the cost and potential harm, acquiring large medical
datasets necessary for deep learning can be difficult. Medical image synthesis
could help mitigate this problem. However, until now, the availability of GANs
capable of synthesizing longitudinal volumetric data has been limited. To
address this, we use the recent advances in latent space-based image editing
to propose a novel joint learning scheme to explicitly embed temporal
dependencies in the latent space of GANs. This, in contrast to previous
methods, allows us to synthesize continuous, smooth, and high-quality
longitudinal volumetric data with limited supervision. We show the
effectiveness of our approach on three datasets containing different
longitudinal dependencies. Namely, modeling a simple image transformation,
breathing motion, and tumor regression, all while showing minimal
disentanglement. The implementation is made available
online111https://github.com/julschoen/Temp-GAN.

###### Keywords:

Generative Adversarial Networks Temporal Generation Semantic Editing.

##  1 Introduction

The use of deep learning in the medical domain has increased recently but is
limited by the need for large and well-labeled datasets [14]. A potential
mitigation to this problem is the use of synthetic data obtained using
generative models such as Generative Adversarial Networks [7], which has been
shown to enhance medical deep learning algorithms [18]. Due to the natural
temporal development of, e.g., disease progression or treatment monitoring,
temporal data is gathered frequently in the medical domain. Prominent cases
are neurodegenerative diseases such as Alzheimer’s or cancer-related
longitudinal data collected during radiotherapy. Combining longitudinal
medical data and deep learning can allow for earlier and more accurate
prognosis [21], as well as disease modeling, such as tumor progression or
regression [24]. GAN have successfully been used to generate temporal data.
They have shown promising results in video generation [17, 20], precipitation
forecasting [16], and also medical temporal data generation [5, 1]. However,
all previous approaches have been either on 2D data [17, 20, 16, 1] or have
considered image-to-sequence or sequence-to-sequence generation tasks [20, 16,
5]. While temporal data generation can be done with image-to-sequence or
sequence-to-sequence models, they do not allow for the generation of new
sequences but rely on input data on which the generated data expands.
In recent years, a line of work has focused on the interpretability of GAN by
investigating them utilizing linear directions in their latent spaces that
result in meaningful and interpretable image transformations [6, 10, 19].
These works show that simple shifts of latent codes along a linear direction
can result in powerful image transformations such as increasing memorability
[6], rotating the image subject [10], or even background removal [19].
However, these approaches operate on pre-trained GAN and can only discover
what is already captured by the learned representation.
The following summarises the main contributions of our work:

  * •

We propose a novel model, jointly training a GAN and a direction in the latent
space corresponding to any desired image transformation for which ordered data
is available. To the best of our knowledge, our approach is the first to
explicitly embed image transformations in the form of linear directions into
the GAN latent space during training. Furthermore, the proposed joint training
procedure is model agnostic and works with any latent variable-based GAN.

  * •

We use the proposed framework to embed a linear direction corresponding to
temporal changes in volumetric medical data. This allows the generation of
longitudinal volumetric data for the first time without requiring input to the
generator in the form of images or sequences of images. Furthermore, as the
temporal sequence generation is based on a simple shift in the latent space,
we can generate smooth and continuous sequences processing each time point
individually, thereby lessening memory requirements compared to the processing
of full sequences.

##  2 Related Work

Our work considers concepts from different lines of prior research in
generative modeling. In the following, we summarise this.

###  2.0.1 Volumetric Data Synthesis

Despite the advances in natural image synthesis, there is a lack of usage and
implementations of general-purpose state-of-the-art volumetric GAN
architectures. Existing volumetric GAN either do not utilize current advances
in GAN architectures [22], are task-specific [12], trade-off image resolution
to allow for state-of-the-art model architectures [8] or are focusing on
image-to-image generation tasks [13, 11]. More advanced architectures, such as
Self-Attention Generative Adversarial Network (SA-GAN) [23], which can be made
more memory efficient by using residual blocks with bottlenecks as suggested
with Big Generative Adversarial Network (BigGAN) [3], are well suited to
volumetric data synthesis. However, while their use is common in the natural
image domain, they are generally not used for volumetric data.

###  2.0.2 Latent Based Image Editing

Latent-based image edits have been possible in GAN latent spaces since the
introduction of Deep Convolutional Generative Adversarial Network (DCGAN)
[15]. Currently, most approaches use linear directions, i.e., latent walks, in
the latent space of pre-trained generators corresponding to interpretable
image edits [6, 19, 10]. The learned representation, however, does not
necessarily contain any potentially desired image transformation. InfoGAN [4]
mitigates this by jointly training the GAN and an additional latent vector
input to disentangle the learned representations. However, a potentially
desired image transformation can not be explicitly enforced. In contrast,
thanks to jointly training the generator and the desired embedding, we ensure
that the desired edit is encoded in the latent space.

###  2.0.3 Temporal Synthesis

Prior works on temporal synthesis have focused on video generation with GAN
[17, 20, 16]. This work was inspired by the shown viability of temporal latent
embedding [17] and the use of two discriminators [17, 20, 16]. However, most
approaches use generators conditioned on input images [20, 16], limiting their
ability to generate new sequences, the exception being Saito et al. [17],
which shows the viability of temporal generation based on latent models, they
do so on natural images and expand the latent space for the temporal modeling.
In contrast, our approach is the first to operate on volumetric data, and we
make this possible by embedding the temporal component into the latent space
directly rather than expanding it.
More similar approaches to ours have been introduced with TR-GAN [5] and the
work by Abdullah et al. [1]. TR-GAN explores many-to-many predictions of
Magnetic Resonance Imaging (MRI) using a single generator. Like the previous
methods, TR-GAN utilizes a generator conditioned on input sequences to predict
temporal sequences. Thus, future time steps are directly generated from input.
In contrast, our approach embeds temporal dependencies in the latent space.
Thus, TR-GAN relies on a sequential generator and cannot generate new
sequences. Finally, Abdullah et al. [1] propose subdividing the latent space
to embed temporal dependencies of medical images. However, they rely on 2D
data, crops around the region of interest, and a-priori information on the
time-dependent variable (e.g., accelerometer data). In contrast, our proposed
method only requires the natural ordering of the temporal sequence, operates
on entire volumes, and we choose linear latent embeddings.

##  3 Methods

In this section, we introduce the proposed framework. Figure 1 shows a
schematic overview of our proposed model architecture.

Figure 1: Schematic overview of our proposed architecture for explicit
embedding in GAN latent space. $z$ is a latent vector, $\boldsymbol{\alpha}$
is a set of shift magnitudes, $z_{1,...,t}$ are the shifted latent vectors,
$x_{1,...,t}$ are shifted images corresponding to synthesized or real data,
$x_{y}$ is the time step corresponding to the original latent vector $z$ if
generated or one real image otherwise, and $x_{i,j,k}$ are three time steps,
where $i,j,k\in\\{1,...,t\\}$.

Our base architecture takes inspiration from video GAN and uses two
discriminators. $D_{im}$ takes individual volumes, i.e., time steps, and
discriminates between real and synthesized data. Thus given an underlying GAN
architecture, the discriminator can be used as $D_{im}$ without further
changes. Next, the architecture has a temporal discriminator $D_{temp}$,
which, given three volumes, discriminates whether they are temporally
consistent. Again, this is implemented using an underlying GAN architecture
and tripling the input channels to allow for the input of temporal sequences.
Further, we use two generators. $G_{im}$ is a traditional generator from a GAN
taking some latent code $z\in\mathbb{R}^{L}$, where $L$ is the latent space
size, and mapping it to an image $x$ without any further changes. Finally, the
temporal generator $G_{temp}$ takes a latent code $z$ and shift magnitudes
$\alpha$ and shifts $z$ by magnitudes $\alpha$ along a learned linear
direction $d\in\mathbb{R}^{L}$ where $L$ is the size of the latent space.
These shifted latent codes are individually used as input to $G_{im}$ to
generate the sequence of volumes. Thus, rather than directly generating a
sequence, we embed a direction in the latent space corresponding to the
desired change, and by shifting with increasing $\alpha$, we can create a set
of latent codes $\vec{z}$ corresponding to consecutive time steps of variable
length. These latent codes are individually used as input to $G_{im}$ to
create the desired number of time steps in data space.
Given the design of the proposed model, any GAN architecture consisting of
discriminator $D$ and generator $G$ can be used by adding the temporal
generator $G_{temp}$ as detailed above and using the discriminator
architecture twice as $D_{im}$ and $D_{temp}$ to construct an explicitly
embedding GAN.
We suggest the following implementation details: Based on the work of Voynov
and Babenko [19], we use a direction $d$ of unit length and
$\alpha\in\mathcal{U}[-6,6]$. As the image discriminator follows standard GAN
training, we suggest using the same loss for the image discriminator and
generator that the base architecture uses.
For temporal consistency, we optimize using adversarial learning. Let
$p_{true}$ be the distribution of real data correctly ordered w.r.t.
transformation magnitude and $p_{false}$ be incorrectly ordered, and $p_{z}$
the latent distribution. Further, let
$\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})$ be any
$\alpha_{i}\in\mathbb{R}$ for which $\alpha_{1}\leq\alpha_{2}\leq\alpha_{3}$,
then we define the adversarial loss objective for the temporal discriminator
using the hinge loss as:

$\begin{split}\underset{D_{temp}}{\min}\>\mathcal{L}_{D_{temp}}&=\underset{x\sim
p_{true}}{\mathbb{E}}[\min(0,1-D_{temp}(x))]\\\ &+\underset{x\sim
p_{false}}{\mathbb{E}}[\min(0,1+D_{temp}(x))]\\\ &+\underset{z\sim
p_{z}}{\mathbb{E}}[\min(0,1+D_{temp}(G_{im}(G_{temp}(z,\boldsymbol{\alpha}))))]\end{split}$
(1)

Given that we want to force the embedding in the latent space, we add a loss
term for both $G_{im}$ and $G_{temp}$ so that $G_{temp}$ learns the direction
and $G_{im}$ learns the latent representation corresponding to the desired
transformation. Thus, we get:

$\underset{G_{im},G_{temp}}{\min}\>\mathcal{L}_{G}=\underset{z,z^{\prime}\sim
p_{z}}{\mathbb{E}}\big{[}-D_{temp}(G_{im}(G_{temp}(z,\boldsymbol{\alpha})))+L_{GAN}(D_{im}(G_{im}(z^{\prime})))\big{]}$
(2)

where $L_{GAN}$ is the applicable GAN loss of the base architecture, and
$\boldsymbol{\alpha}$ and $p_{z}$ are defined as above.
Intuitively, the temporal discriminator learns to discriminate based on the
transformation we aim to embed. Therefore, the generators trying to fool the
temporal discriminator must generate data that exhibits the correct
transformation and does not change the scene (e.g., patient) markedly.
We evaluate image quality using visual inspection and slice-wise Fréchet
Inception Distance (FID) and visual inspection for temporal consistency. The
shifted images should not be of worse image quality than images resulting from
directly sampled latent codes. Thus, the basic image quality measures are also
used to assess the shifted images.

##  4 Experiments

###  4.0.1 Datasets

We evaluate the proposed architecture on three volumetric thoracic
Computerized Tomography (CT) datasets.

  * •

The Lung Image Database Consortium image collection (LIDC) [2]. We preprocess
LIDC by limiting the intensity range to $[-1000,2000]$ Hounsfield Units (HU)
and normalize to a range of $[-1,1]$ using min-max scaling. We resize the data
to $128\times 128\times 128$ voxels to limit computational demands. To test
the approach, we introduce a simple image transformation corresponding to
shifts along the $x$-axis with a shift magnitude of $[-32,32]$ voxels randomly
selected.

  * •

Breathing Motion. The dataset, collected at Rigshospitalet, Copenhagen,
Denmark, consists of longitudinal thoracic CT scans showing the breathing
phases of $499$ non-small cell lung cancer patients treated with radiotherapy
between $2009$ to $2017$. Each data point generally consists of $10$ scans,
where the first five correspond to exhaling and the following five to
inhaling. We only consider the scans corresponding to exhaling. We limit the
range to $[-1000,500]$ HU and normalize to a range of $[-1,1]$ using min-max
scaling. We resize the data to $128\times 128\times 64$ voxels to limit
computational demands.

  * •

Tumor Regression. The final dataset consists of $256$ patients, a subset of
the patients from the Breathing Motion dataset, for which at least $10$ daily
treatment thoracic cone beam CT scans were available. It shows tumor
regression during radiotherapy. We apply the same preprocessing as with the
breathing motion dataset.

For all three datasets, we use $90\%$ of the patients for training and $10\%$
for testing. Figure 2 shows an example of the breathing motion and tumor
regression dataset.

Figure 2: Examples of breathing motion and tumor regression datasets. The
presented examples are after preprocessing. For breathing motion, the center
volume shows the most exhaled while the ones to the left correspond to
exhaling and those to the right to inhaling. For tumor regression, we estimate
the slice corresponding to the center of the tumor manually. The tumor is
marked with the red bounding box.

###  4.0.2 Implementation Details

We run all experiments on two Nvidia RTX A6000. We use Python $3.9$ and
PyTorch $1.11.0$ for the implementation. The first author performed all visual
inspections without formal education in medical image evaluation. We adapt SA-
GAN [23] to volumetric data and use it as our base GAN architecture following
the parameters (e.g., learning rate and optimizer) suggested by the authors
unless otherwise specified. Throughout all experiments, we sample
$\alpha\sim\mathcal{U}[-6,6]$ with a linear direction $d$ of unit length based
on Voynov and Babenko [19], and we arbitrarily choose to train for $5000$
iterations. We use a batch size of $8$ for the LIDC dataset and a batch size
of $16$ for the other two to fit the memory of the used GPUs. For the LIDC
dataset, we use a latent space size of $L=512$ and reduce it to $L=256$ for
the other two, as the resolution of the datasets is halved as well.

##  5 Results

We present the final, unbiased estimate of the image quality of the proposed
model on all three datasets in Table 1. Note that some image transformations
are more readily visible in video format. Therefore, consider the generated
volumes as videos in the provided GitHub repository.

|  FID ax. |  FID cor. |  FID sag.
---|---|---|---
LIDC | $93.8\pm 1.0$ | $54.3\pm 0.5$ | $30.0\pm 1.2$
Breathing Motion | $139.2\pm 2.7$ | $79.8\pm 1.0$ | $99.6\pm 1.4$
Tumor Regression | $82.2\pm 2.7$ | $42.3\pm 1.4$ | $53.4\pm 0.8$
Table 1: Image Quality of the temporal GAN trained on all three datasets. All
models are trained on $90\%$ of the scans split patient-wise and evaluated on
$10\%$. The FID scores are calculated using random time steps of real and
synthesized data.

Examples of generated volumes of the model trained on the full LIDC data set
are given in Figure 3.

Figure 3: Four examples of generated volumes with embedded shift for the
proposed model on the LIDC dataset. The center volume corresponds to the
original latent vector. All images show the center slice for the sagittal,
coronal, and axial view.

The resulting volumes are of high quality, and we observe the desired image
transformation embedded as a shift in the latent space. The transition between
shifted images is smooth with minimal entanglement.
Next, we consider the breathing motion dataset. Figure 4 presents some
generated volumes for the breathing motion dataset.

Figure 4: Four examples of generated volumes with embedded shift for the
proposed model on the breathing motion dataset. The center volume corresponds
to the original latent vector. All images show the center slice for the
sagittal, coronal, and axial view.

We observe good image quality with sufficient detail and realistic anatomy.
Considering the embedding, we observe that breathing motion is captured well,
and the temporal dependencies of breathing are embedded in the latent space.
The clearest change when moving along the embedded direction is the diaphragm
moving upward while exhaling. This is also the most obvious change observable
in the real data. Additionally, we observe the stomach or rib cage contracting
while exhaling. Lastly, we observe very high scene consistency. I.e., the
generated scan of the patient does not change markedly while moving the latent
code along the embedded direction. Thus, the patient’s anatomy is preserved,
and the changes induced by moving along the embedded direction are restricted
to breathing-related changes.
Finally, we consider the model trained on the tumor regression dataset. We
present examples of generated volumes in Figure 5.

Figure 5: Four examples of generated volumes with embedded shift for the
proposed model on the tumor regression dataset. The center volume corresponds
to the original latent vector. For all images, we try to locate the center of
the tumor for the sagittal, coronal, and axial view.

The image quality of the generated volumes is good, showing details in the
vessel, tissue, and bone structure. We observe volumes both containing and not
containing tumors. Those generated volumes containing tumors have them in
varied places, shapes, and sizes. If tumors are present in the generated
volumes, the temporal generation results in tumors shrinking in size. I.e.,
the model successfully embeds temporal tumor regression in image space as a
linear direction in the latent space. When traversing the embedded direction,
there is minimal change to the volumes other than the reduction in tumor size.
Further, no clear change exists if no tumor is present in the volume. Thus,
the direction models tumor regression in a disentangled manner.

##  6 Discussion

###  6.0.1 Temporal Generation

We can use a simple learned direction to generate temporal sequences of data
using only a single non-temporal generator, which, to the best of our
knowledge, has not been shown previously. Our proposed method of jointly
training such a direction and the GAN shows distinct benefits over discovering
directions in pretrained generators. From visible inspection, our method shows
almost no entanglement and ensures that even complex transformations, such as
tumor regression, are enforced to be present as a linear direction in the
latent space.
Our model shows high-quality synthetic data with controllable enforced image
transformations with smooth continuous generation on three datasets.
Additionally, in particular, on the breathing motion and tumor regression
dataset, we observe clearer changes than observed in the data (e.g., movement
of the diaphragm in the breathing motion dataset). This indicates that the
proposed method isolates the signal corresponding to temporal development
well. Compared to other temporal generation approaches, our model does not
require conditioning of the generator or a sequential generator. As a
consequence, we can easily vary the architecture and benefit from advances in
GAN architectures that are to come.
The results we observe on the tumor regression dataset deserve the most
attention. While traditional tumor regression models might be more patient-
and therapy-specific [9], the scale we operate on is novel. Furthermore,
unlike previous methods, we generate entire volumes and show that we can model
tumor regression as part of the image generation process.

###  6.0.2 Limitations

We use the parameters used for SA-GAN. While this is likely a good choice for
the image generation component of our proposed model, the temporal aspects
could perform better with different parameters and architecture choices.
Moreover, as there is little prior investigation into evaluating temporal GAN,
we needed to devise our evaluation strategy, further development in this area
would likely be beneficial.

###  6.0.3 Impact & Future Work

As our method is trained based only on the order of the transformation
magnitude, it is reasonable to assume that it can be applied to any
transformation where such an ordering exists. Further, our method provides
many practical applications to downstream machine learning tasks by providing
a method to synthesize controllable data, e.g., with or without tumor, in a
domain where annotations are costly and difficult to obtain. We provide a
proof of concept of unsupervised tumor segmentation using our method in Figure
6.

Figure 6: Proof of concept of unsupervised tumor segmentation using our
method. We generate a future time point to a given volume, take the difference
image, and threshold it with a threshold of $0.2$. Then we apply two erosion
followed by two dilation operations to remove noise and are left with the
segmentation mask. Next to the direct application to tumor segmentation, the
difference image might also visualize treatment effects, such as weight loss,
which is a common side effect.

Further practical applications directly benefiting medical image analysis will
likely arise with further investigation of our method. Given the clear and
isolated signal we observe, natural applications of our work could be in the
visualization and understanding of changes happening in temporal sequences.
Additionally, investigating how much the sampled temporal development reflects
patient-specific as opposed to therapy-specific aspects would offer valuable
insights. Finally, we see the future investigation of embedding non-temporal
dependencies as one of the most promising potential applications. Embedding
transformations across patients, e.g., disease severity, would allow for
further fine-grained control over data synthesis in the medical domain.

##  7 Conclusion

In this work, we investigate the possibility of explicitly embedding temporal
dependencies in the latent space of generative models to generate longitudinal
volumetric data. We generate controllable longitudinal data with minimal
entanglement. Further, linear directions in the latent space are sufficient to
generate temporal sequences from a non-temporal generator. Due to the
simplicity of the linear latent walk, we can generate continuous and smooth
sequences of varying lengths, unlike other suggested temporal GAN.
We show that our framework can generate complex temporal dependencies, e.g.,
breathing motion or tumor regression, as part of the image synthesis task on
medical data. The method could potentially improve unsupervised tumor
segmentation, disease-aware image augmentation, and radiotherapy planning.
Further, our method can embed any temporal dependency with limited supervision
and thus provides further usefulness beyond what we explore in this work.

###  7.0.1 Acknowledgements

The authors acknowledge the National Cancer Institute and the Foundation for
the National Institutes of Health, and their critical role in the creation of
the free publicly available LIDC/IDRI Database used in this study. The authors
would like to thank Anna Kirchner for help in preparation of the manuscript.
Jens Petersen is partly funded by research grants from the Danish Cancer
Society (grant no. R231-A13976) and Varian Medical Systems.

## References

  * [1] Abdullah, Holler, M., Kunisch, K., Landman, M.S.: Nonlinear motion separation via untrained generator networks with disentangled latent space variables and applications to cardiac MRI. arXiv abs/2205.10367 (2022)
  * [2] Armato III, S.G., McLennan, G., Bidaut, L., McNitt-Gray, M.F., Meyer, C.R., Reeves, A.P., Zhao, B., Aberle, D.R., Henschke, C.I., Hoffman, E.A., et al.: The lung image database consortium (lidc) and image database resource initiative (idri): A completed reference database of lung nodules on ct scans. Medical physics 38(2), 915–931 (2011)
  * [3] Brock, A., Donahue, J., Simonyan, K.: Large scale GAN training for high fidelity natural image synthesis. In: 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net (2019)
  * [4] Chen, X., Duan, Y., Houthooft, R., Schulman, J., Sutskever, I., Abbeel, P.: Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In: Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain. pp. 2172–2180 (2016)
  * [5] Fan, C.C., Peng, L., Wang, T., Yang, H., Zhou, X.H., Ni, Z.L., Wang, G., Chen, S., Zhou, Y.J., Hou, Z.G.: Tr-gan: Multi-session future mri prediction with temporal recurrent generative adversarial network. IEEE Transactions on Medical Imaging pp. 1–1 (2022)
  * [6] Goetschalckx, L., Andonian, A., Oliva, A., Isola, P.: Ganalyze: Toward visual definitions of cognitive image properties. In: 2019 IEEE/CVF International Conference on Computer Vision, ICCV 2019, Seoul, Korea (South), October 27 \- November 2, 2019. pp. 5743–5752. IEEE (2019)
  * [7] Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. In: Advances in Neural Information Processing Systems. vol. 27. Curran Associates, Inc. (2014)
  * [8] Hong, S., Marinescu, R.V., Dalca, A.V., Bonkhoff, A.K., Bretzner, M., Rost, N.S., Golland, P.: 3d-stylegan: A style-based generative adversarial network for generative modeling of three-dimensional medical images. In: Deep Generative Models, and Data Augmentation, Labelling, and Imperfections - First Workshop, DGM4MICCAI 2021, and First Workshop, DALI 2021, Held in Conjunction with MICCAI 2021, Strasbourg, France, October 1, 2021, Proceedings. Lecture Notes in Computer Science, vol. 13003, pp. 24–34. Springer (2021)
  * [9] Huang, Z., Mayr, N.A., Yuh, W.T., Lo, S.S., Montebello, J.F., Grecula, J.C., Lu, L., Li, K., Zhang, H., Gupta, N., Wang, J.Z.: Predicting Outcomes in Cervical Cancer: A Kinetic Model of Tumor Regression during Radiation Therapy. Cancer Research 70(2), 463–470 (2010)
  * [10] Jahanian, A., Chai, L., Isola, P.: On the "steerability" of generative adversarial networks. In: 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net (2020)
  * [11] Lan, H., Initiative, A.D.N., Toga, A.W., Sepehrband, F.: Three-dimensional self-attention conditional gan with spectral normalization for multimodal neuroimaging synthesis. Magnetic Resonance in Medicine 86(3), 1718–1733 (2021)
  * [12] Li, F., Huang, W., Luo, M., Zhang, P., Zha, Y.: A new VAE-GAN model to synthesize arterial spin labeling images from structural MRI. Displays 70, 102079 (2021)
  * [13] Lin, W.: Synthesizing missing data using 3d reversible GAN for alzheimer’s disease. In: ISAIMS 2020: International Symposium on Artificial Intelligence in Medical Sciences, Beijing, China, September, 2020. pp. 208–213. ACM (2020)
  * [14] Litjens, G., Kooi, T., Bejnordi, B.E., Setio, A.A.A., Ciompi, F., Ghafoorian, M., van der Laak, J.A.W.M., van Ginneken, B., Sánchez, C.I.: A survey on deep learning in medical image analysis. Medical Image Anal. 42, 60–88 (2017)
  * [15] Radford, A., Metz, L., Chintala, S.: Unsupervised representation learning with deep convolutional generative adversarial networks. In: 4th International Conference on Learning Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings (2016)
  * [16] Ravuri, S., Lenc, K., Willson, M., Kangin, D., Lam, R., Mirowski, P., Fitzsimons, M., Athanassiadou, M., Kashem, S., Madge, S., et al.: Skilful precipitation nowcasting using deep generative models of radar. Nature 597(7878), 672–677 (2021)
  * [17] Saito, M., Matsumoto, E., Saito, S.: Temporal generative adversarial nets with singular value clipping. In: IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017. pp. 2849–2858. IEEE Computer Society (2017)
  * [18] Sandfort, V., Yan, K., Pickhardt, P.J., Summers, R.M.: Data augmentation using generative adversarial networks (CycleGAN) to improve generalizability in CT segmentation tasks. Scientific Reports 9, 16884 (2019)
  * [19] Voynov, A., Babenko, A.: Unsupervised discovery of interpretable directions in the GAN latent space. In: Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event. Proceedings of Machine Learning Research, vol. 119, pp. 9786–9796. PMLR (2020)
  * [20] Wang, Y., Bilinski, P., Brémond, F., Dantcheva, A.: Imaginator: Conditional spatio-temporal GAN for video generation. In: IEEE Winter Conference on Applications of Computer Vision, WACV 2020, Snowmass Village, CO, USA, March 1-5, 2020. pp. 1149–1158. IEEE (2020)
  * [21] Wen, J., Thibeau-Sutre, E., Diaz-Melo, M., Samper-González, J., Routier, A., Bottani, S., Dormont, D., Durrleman, S., Burgos, N., Colliot, O.: Convolutional neural networks for classification of alzheimer’s disease: Overview and reproducible evaluation. Medical Image Anal. 63, 101694 (2020)
  * [22] Wu, J., Zhang, C., Xue, T., Freeman, B., Tenenbaum, J.: Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling. In: Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain. pp. 82–90 (2016)
  * [23] Zhang, H., Goodfellow, I.J., Metaxas, D.N., Odena, A.: Self-attention generative adversarial networks. In: Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA. Proceedings of Machine Learning Research, vol. 97, pp. 7354–7363. PMLR (2019)
  * [24] Zhu, H.B., Xu, D., Ye, M., Sun, L., Zhang, X.Y., Li, X.T., Nie, P., Xing, B.C., Sun, Y.S.: Deep learning-assisted magnetic resonance imaging prediction of tumor response to chemotherapy in patients with colorectal liver metastases. International Journal of Cancer 148(7), 1717–1730 (2021)

  *[GAN]: Generative Adversarial Network
  *[SA-GAN]: Self-Attention Generative Adversarial Network
  *[BigGAN]: Big Generative Adversarial Network
  *[DCGAN]: Deep Convolutional Generative Adversarial Network
  *[MRI]: Magnetic Resonance Imaging
  *[FID]: Fréchet Inception Distance
  *[CT]: Computerized Tomography
  *[LIDC]: The Lung Image Database Consortium image collection
  *[HU]: Hounsfield Units


# Van der Waals hetrostructure configuration effect on exciton and
thermoelectric characteristics

T.E. Ada  K.N. Nigussa<EMAIL_ADDRESS>Department of Physics, Dilla
University, P.O. Box 419, Dilla, Ethiopia  Department of Physics, Addis Ababa
University, P.O. Box 1176, Addis Ababa, Ethiopia

###### Abstract

A GW calculation based on a truncated coulomb interaction with an added small
q limit was applied to 2D van der Waals hetro-layered structures, and the Kane
dispersion model was used to determine the accurate band gap edge. All ab
initio calculations were performed with the gpaw package. Our findings show
that layering the same or different types of atoms with a vacuum in between
has an enormous impact on band alignment, effective mass of holes and
electrons, exciton binding energy, thermoelectric characteristics, density of
hot electrons, and electronic band gaps.Thus, layered interactions of the same
kind constrain the configuration to have a direct band gap, whereas different
kinds allow for an indirect gap, resulting in an indirect exciton with a
greater binding energy due to the band confinement effect. Also, the hetro
structure of the utmost configuration allows hot electrons to relax by
creating unoccupied states; as a result, a high density of hot electrons
distribute themselves over unoccupied states, making it less sensitive to
temperature; in other words, electron-hole pairing lasts longer due to thermal
drop. Therefore, among several possible configurations,
$\mathrm{MoSe_{2}-MoS_{2}}$ and $\mathrm{MoSeSe-MoSSe}$ have an enhanced
optical gap plus multiple excitation peaks that is similar to the experimental
photoluminescence spectrum, as well as an improved seebeck coefficient due to
low thermal conductivity and the ability to generate a higher density of
states and empty states.

###### keywords:

2D van der Waals heterostructure , Effective mass, Thermoelectric materials,
Exciton, Seebeck coefficient, Parabolic constant , Kane dispersion equation.

##  1 Introduction

2D van der Waals heterostructures are a preferred choice for high-efficiency
solar devices. A 2D material can exhibit remarkable features, particularly in
optics and electron transport, due to strong covelent bonding that provides
in-plane stability and van der Waals forces that bind the stack together, as
well as electron-hole pairing in adjacent layers.

Interlayer electron-hole pairing occurs in a 2D van der Waals heterostructure
when there is free space between the bottom of the conduction band and the top
of the valence band, and forces take place between atoms or electron density
sections.Thus, interlayer electron-hole pairing in varied configurations is
expected to profoundly impact photon absorption and optical characteristics in
solar cells. Hot electrons released during solar cell heating, on the other
hand, have the ability to affect optical characteristics across multiple
layers.

However, a thorough knowledge of interlayer electron-hole pairing and hot
electrons in relation to the configuration of a 2D van der Waals
heterostructure is still lacking due to inaccurate estimates of band gap and
band aligment at interfaces [1].

Density functional theory calculations are notoriously challenging for
determining band gaps and band alignment at interfaces of 2D semiconductor .
To get reliable band energies, many-body perturbation theory, such as the GW
approximation, and the Bethe Salpeter Equation (BSE) should be used, with
excition effects incorporated [2]. However, the computational expense of such
approaches renders them inappropriate for van der Waals hetrostructures
containing more than a few stacked layer.

Here we carried out a $\mathrm{G_{0}W_{0}}$ calculation of 2D truncated
coulomb interaction with added analytic correction for the small $\mathrm{q}$
limit of less dense k-point grid convergence [3], and To circumvent band
alignment problems, we employed the parabolic-band approximation, which breaks
down at high photon (electron) energies and can be incorporated in an
approximate way similar to Ridley’s method for hot-electron transport [4, 5].

We explore the interlayer electron-hole pairing and hot electrons of a van der
Waals heterostructure of a 2H-bilayer ($\mathrm{MoS_{2}|MoSe_{2}}$)(see Fig 1)
in various configurations of the highest symmetry point of the first Brillouin
zone, which is where the lowest bound exciton is situated(see Fig 2).

(a) $\mathrm{MoS_{2}-MoS_{2}}~{}(a=3.17~{}\AA)$

(b) $\mathrm{MoSeS-MoS_{2}}~{}(a=3.21~{}\AA)$

(c) $\mathrm{MoSe_{2}-MoS_{2}}~{}(a=3.25~{}\AA)$

(d) $\mathrm{MoSe_{2}-MoSe_{2}}~{}(a=3.32~{}\AA)$

(e) $\mathrm{MoSSe-MoSe_{2}}~{}(a=3.28~{}\AA)$

(f) $\mathrm{MoSSe-MoSeS}~{}(a=3.25~{}\AA)$

(g) $\mathrm{MoSeS-MoSSe}~{}(a=3.25~{}\AA)$

(h) $\mathrm{MoSSe-MoSSe}~{}(a=3.25~{}\AA)$

Figure 1:  Possible hetrostructured stacking combinations of
$\mathrm{MoS_{2}}$ and $\mathrm{MoSe_{2}}$ colored by constituent elements:
yellow for Sulfur, gray for Molybdenum, and green for Selenium. For
interpretation of color code references, refer to the web version of this
article.

##  2 Computational method

All the ab-initio calculation were perfomed using gpaw code [6], and projected
augmented wave is used for the Kohn-Sham self-consistant calculations [7, 8].
The wave functions are expanded in a plane wave cut-off energy below 600 eV
with double-zeta polarized atomic orbitals basis set [9]. The $k$-points
within Brillouin zone (BZ) is chosen according to Monkhorst-Pack scheme [10],
where a $\rm\bf{k}$ meshes of 18$\times$18$\times$2\. The interactions of the
valence electrons with the core electrons and nuclei is treated using an
approximation within a projector augmented wave (paw) data sets [9, 11].
Geometry optimizations of the atomic coordinates and the unit cell degrees of
freedom is done with implementation of the stress tensor [12, 13, 14, 15]. The
convergence criteria for the forces were set at 0.005 eV/Å. The exchange-
correlation energies are approximated within the generalized gradient
approximation of PBE [16] $\&$ a k mesh of 2$\times$2$\times$1 is used in a
geometry relaxation calculations, and applied to 2H-heterostructure consists
of two mono layers with 2.0 $\AA$ vacuum in between, and the surface is
insulated from external interaction by 10 $\AA$ vacuum along the c-axis
direction.

In the calculations of the bandgaps, we have implemented an exchange-
correlation functionals GLLBSC, $\mathrm{vdW-DF2}$ and PBE within GW
approximations [17, 18]. Calculation of density of states (DOS) $\&$ the group
velocity [19], $\mathrm{v(\varepsilon)}$ are done from a graph of
$\varepsilon\rm_{k}$ versus $k$, and DOS at a particular energies
$\varepsilon$ is given as,

${}\mathrm{N({\varepsilon})=\sum\limits_{N}w\rm_{N}{\delta}({\varepsilon}-{\varepsilon\rm_{N}})},$
(1)

and,

$\mathrm{v({\varepsilon})=\frac{k^{2}}{2\pi^{2}}\frac{1}{N({\varepsilon})\hbar}}$
(2)

where $\mathrm{N=(k,s)}$ is an occupation state corresponding to a
$\mathrm{k}$ point $\&$ a spin $s$, $\&$ $\mathrm{w\rm_{N}}$ is a weight
factor respectively.

For better match to the experimental results, we used $\mathrm{vdW-DF2}$ [20,
21, 22] for layer interaction and GLLB-SC for London force exclusion to
distinguish the van der Waals interaction force effect.

We used parabolic band approximation to study the influence of band alignment
at interfaces, and the $\mathrm{E(k)}$ relation may be stated directly from
Kane’s theory [23], as is often used in literature by

$\mathrm{\frac{\hbar^{2}k^{2}}{2m^{*}}=E(1+\alpha E),\text{~{}~{}where,
}\alpha\approx\frac{1}{E_{g}}}$ (3)

Where $\mathrm{m^{*}}$ is the effective mass at the band edge,
$\mathrm{E_{g}}$ is the band gap, and the energy $\mathrm{E}$ is measured from
the band edge.

##  3 Results and discussion

The optimized lattice constants for $\mathrm{2H-MoS_{2}}$ and
$\mathrm{2H-MoSe_{2}}$ are $3.17~{}\AA$ and $3.32~{}\AA$, respectively, as
shown in Fig. 1, which are impressively similar to the experimental values of
$3.18~{}\AA$ and $3.33~{}\AA$ in the order reported in Ref [24]. Obviously,
the rest structure lattice constant falls between (3.17-3.33).

Fig. 2 illustrates that a quasi-particle band gap estimated using the Bethe
Salpeter Equation (BSE) is equal to the sum of the optical gap and the exciton
binding energy, and is a popular choice because it produces the exact band
gap. The $\mathrm{G_{0}W_{0}}$ [25], on the other hand, yielded somewhat
similar band gap values. One can note that
$\mathrm{E_{B}+E_{g}^{opt}=E_{g}^{GW}-E_{g}^{KS}}$, exciton binding energy and
optical gap are often difficult to distinguish due to a lack of a distinct
line between photoabsorption and resonant excitation energy. Thus, the
estimated values of exciton binding energy for $\mathrm{2H-MoS_{2}}$ and
$\mathrm{2H-MoSe_{2}}$ are $\mathrm{1.99~{}eV}$ and $\mathrm{1.68~{}eV}$,
respectively, which are surprisingly similar to the experimental values within
the range of $\mathrm{1.96-1.99~{}eV}$ and $\mathrm{1.682-1.712~{}eV}$ in the
corresponding order as published in Refs [26, 27, 28, 29]. As a result, we can
conclude that the remaining structural configuration of
$\mathrm{E_{B}+E_{g}^{opt}}$ will lie between the two ranges specified above.

Figure 2:  Fundamental gap, $\mathrm{E_{g}^{BSE}}$ could constitute optical
gap, $\mathrm{E_{g}^{opt}}$, the electron-hole or excitonic binding energy,
$\mathrm{E_{B}}$, while GW HOMO-LUMO gaps, $\mathrm{E_{g}^{GW}}$ and
$\mathrm{E_{g}^{GLLB-SC}}$ consisting of electronic gap, $\mathrm{E_{g}^{KS}}$
and band discontinuity, $\mathrm{\Delta_{xc}}$. For interpretation of color
code references, refer to the web version of this article. Figure 3:  In
response to band edge energy, $\mathrm{E=\frac{1}{\alpha}}$, the band
alignment of hetrostructural configuration initiates either direct or indirect
electron and hole binding.

In two-dimensional electronic systems, the coulomb interaction between
electrons and holes is much stronger. Band structure effects are incorporated
using an excitonic effective mass and dielectric screen. Exciton binding
energies are directly proportional to band gaps. Fig. 3 depicts how excitonic
effects bound at different and identical k-points, resulting in changes in
exciton gap size. Mono and bilayers have a higher amount due to their lower
dielectric constants [30].

Figs. 4(a) and 4(b) highlight that the MoSSe-MoSSe and MoSe2-MoS2
hetrostructure configurations have a higher number of hot-electron
populations, giving rise to a gradual increase in hot electron mobility.
However, when temperature rises, the hot-electron population gradually
declines, except in the aforementioned structure, triggering relatively fast
hot electron mobility. Therefore, $\mathrm{MoSe_{2}-MoS_{2}}$ and
$\mathrm{MoSeSe-MoSSe}$ were positioned at the maximum achievable hetrocity,
resulting in increased hot electron density and a significant contribution to
the steady increase in hot electron mobility.

(a) The detrimental effect of layer configuration on hot electron density as
temperature changes. For the interpretations of the references to color in
this plot legend, the reader is referred to the web-version.

(b) Hot electron velocity rises in response to temperature changes in their
respective layer configurations. For the interpretations of the references to
color in this plot legend, the reader is referred to the web version.

Figure 5:  The equation of state calculation for $\mathrm{MoS_{2}-MoS_{2}}$
using the parabolic fit (green broken line) and Murnaghan fit (solid red line)
compared to $\mathrm{MoSe_{2}-MoSe_{2}}$ on the right.

The equation of state calculation for $\mathrm{MoS_{2}-MoS_{2}}$ using the
parabolic fit (green broken line) and Murnaghan fit (solid red line) compared
to $\mathrm{MoSe_{2}-MoSe_{2}}$ on the right.

A greater bulk modulus number indicates a stronger resistance to high
pressures and compression. Consequently, $\mathrm{MoSe_{2}-MoSe_{2}}$
structures are slightly less resistant to high pressure and compression than
$\mathrm{MoS_{2}-MoS_{2}}$. The bulk modulus values in Table 2 and 3 are
calculated from curve fits using the Murnaghan equation, Eq. (4), as shown in
Fig .5. Hence, some findings are shown in Table 2, such as
$\mathrm{MoSe_{2}-MoSe_{2}}$ being 20.99 GPa, which is consistent with the
experimental value of 20.55 GPa [31].

$E(\eta)=E_{0}+\frac{9B_{0}V_{0}}{16}((\eta^{2}-1)^{2}(6+B^{{}^{\prime}}_{0}(\eta^{2}-1)-4\eta^{2})$
(4)

Where $\eta=\left(\frac{V}{V_{0}}\right)^{\frac{1}{3}}$, $B_{0}$ and
$B^{{}^{\prime}}_{0}$ are the bulk modulus and its pressure derivative at the
equilibrium volume $V_{0}$ [32].

Table 1: Calculations of Effective mass of electron and hole,
$\mathrm{m^{*}}$, Reduced mass, $\mathrm{\mu}$, Parabolic constant,
$\mathrm{\alpha~{}[1/eV]}$, Electron transition on same or different k-point
for $\mathrm{MoS_{2}-MoS_{2}}$, $\mathrm{MoSeS-MoS_{2}}$ ,
$\mathrm{MoSe_{2}-MoS_{2}}$ , $\mathrm{MoSe_{2}-MoSe_{2}}$ , $\&$
$\mathrm{MoSSe-MoSe_{2}}$ with vdW-DF2 and GLLB-SC exchange correlations.

|  | $\mathrm{\underline{vdW-DF2}}$ |  |  |  | $\mathrm{\underline{GLLB-SC}}$ |  |
---|---|---|---|---|---|---|---|---
System | $\mathrm{~{}m^{*}}$ | $\mu$ | $\mathrm{\alpha~{}[1/E_{g}]}$ | $\mathrm{Transition,~{}V~{}\rightarrow~{}C}$ | $\mathrm{m^{*}}$ | $\mu$ | $\mathrm{\alpha~{}[1/E_{g}]}$ | $\mathrm{Transition,~{}V~{}\rightarrow~{}C}$
| $\mathrm{m_{h}=-0.29}$ |  | $\mathrm{CB=-2.41}$ |  | $\mathrm{m_{h}=-0.62}$ |  | $\mathrm{CB=-0.25}$ |
$\mathrm{MoS_{2}-MoS_{2}}$ |  | 0.20 |  | $\mathrm{H~{}\rightarrow~{}K}$ |  | 0.13 |  | $\mathrm{K~{}\rightarrow~{}K}$
| $\mathrm{m_{e}=0.69}$ |  | $\mathrm{VB=0.75}$ |  | $\mathrm{m_{e}=0.28}$ |  | $\mathrm{VB=0.92}$ |
| $\mathrm{m_{h}=-0.26}$ |  | $\mathrm{CB=-3.34}$ |  | $\mathrm{m_{h}=-0.47}$ |  | $\mathrm{CB=-2.03}$ |
$\mathrm{MoSeS-MoS_{2}}$ |  | 0.15 |  | $\mathrm{K~{}\rightarrow~{}K}$ |  | 0.19 |  | $\mathrm{K~{}\rightarrow~{}H}$
| $\mathrm{m_{e}=0.37}$ |  | $\mathrm{VB=0.87}$ |  | $\mathrm{m_{e}=0.33}$ |  | $\mathrm{VB=1.01}$ |
| $\mathrm{m_{h}=-0.24}$ |  | $\mathrm{CB=-2.81}$ |  | $\mathrm{m_{h}=-0.21}$ |  | $\mathrm{CB=-4.43}$ |
$\mathrm{MoSe_{2}-MoS_{2}}$ |  | 0.19 |  | $\mathrm{H~{}\rightarrow~{}K}$ |  | 0.16 |  | $\mathrm{H~{}\rightarrow~{}K}$
| $\mathrm{m_{e}=0.81}$ |  | $\mathrm{VB=3.34}$ |  | $\mathrm{m_{e}=0.73}$ |  | $\mathrm{VB=1.84}$ |
| $\mathrm{m_{h}=-0.32}$ |  | $\mathrm{CB=-1.69}$ |  | $\mathrm{m_{h}=-0.27}$ |  | $\mathrm{CB=-1.13}$ |
$\mathrm{MoSe_{2}-MoSe_{2}}$ |  | 0.22 |  | $\mathrm{H~{}\rightarrow~{}H}$ |  | 0.19 |  | $\mathrm{H~{}\rightarrow~{}K}$
| $\mathrm{m_{e}=0.75}$ |  | $\mathrm{VB=1.17}$ |  | $\mathrm{m_{e}=0.66}$ |  | $\mathrm{VB=1.72}$ |
| $\mathrm{m_{h}=-0.60}$ |  | $\mathrm{CB=-2.96}$ |  | $\mathrm{m_{h}=-0.24}$ |  | $\mathrm{CB=-1.31}$ |
$\mathrm{MoSSe-MoSe_{2}}$ |  | 0.23 |  | $\mathrm{K~{}\rightarrow~{}H}$ |  | 0.14 |  | $\mathrm{K~{}\rightarrow~{}H}$
| $\mathrm{m_{e}=0.37}$ |  | $\mathrm{VB=1.02}$ |  | $\mathrm{m_{e}=0.33}$ |  | $\mathrm{VB=1.72}$ |
| $\mathrm{m_{h}=-0.65}$ |  | $\mathrm{CB=-2.79}$ |  | $\mathrm{m_{h}=-0.27}$ |  | $\mathrm{CB=-4.47}$ |
$\mathrm{MoSSe-MoSeS}$ |  | 0.35 |  | $\mathrm{H~{}\rightarrow~{}H}$ |  | 0.19 |  | $\mathrm{H~{}\rightarrow~{}H}$
| $\mathrm{m_{e}=0.76}$ |  | $\mathrm{VB=0.82}$ |  | $\mathrm{m_{e}=0.66}$ |  | $\mathrm{VB=0.73}$ |
| $\mathrm{m_{h}=-0.30}$ |  | $\mathrm{CB=-3.03}$ |  | $\mathrm{m_{h}=-0.26}$ |  | $\mathrm{CB=-1.14}$ |
$\mathrm{MoSeS-MoSSe}$ |  | 0.16 |  | $\mathrm{K~{}\rightarrow~{}K}$ |  | 0.14 |  | $\mathrm{K~{}\rightarrow~{}K}$
| $\mathrm{m_{e}=0.36}$ |  | $\mathrm{VB=0.80}$ |  | $\mathrm{m_{e}=0.31}$ |  | $\mathrm{VB=1.39}$ |
| $\mathrm{m_{h}=-0.30}$ |  | $\mathrm{CB=-2.38}$ |  | $\mathrm{m_{h}=-0.26}$ |  | $\mathrm{CB=-2.56}$ |
$\mathrm{MoSSe-MoSSe}$ |  | 0.23 |  | $\mathrm{H~{}\rightarrow~{}K}$ |  | 0.19 |  | $\mathrm{H~{}\rightarrow~{}K}$
| $\mathrm{m_{e}=0.92}$ |  | $\mathrm{VB=1.49}$ |  | $\mathrm{m_{e}=0.66}$ |  | $\mathrm{VB=1.24}$ |

Table 2: Calculations of effective mass of hole of parabolic constant,
$\mathrm{\alpha~{}[1/eV]}$ to corresponding energy of band index,
$\mathrm{\varepsilon_{N}}$ $\&$ Bulk modulus, B for
$\mathrm{MoS_{2}-MoS_{2}}$, $\mathrm{MoSeS-MoS_{2}}$ ,
$\mathrm{MoSe_{2}-MoS_{2}}$ , $\mathrm{MoSe_{2}-MoSe_{2}}$ , $\&$
$\mathrm{MoSSe-MoSe_{2}}$ with vdW-DF2 and GLLB-SC exchange correlations plus
PBE within GW approximation.

|  |  | $\mathrm{\underline{vdW-DF2}}$ |  |  |  | $\mathrm{\underline{GLLB-SC}}$ |  |  |  | $\underline{\mathrm{G_{0}W_{0}@PBE}}$
---|---|---|---|---|---|---|---|---|---|---|---
System | $\mathrm{Band}$ | $\mathrm{\varepsilon_{N}~{}[eV]}$ | $\mathrm{m_{h}}$ | $\mathrm{~{}\alpha}$ | B[GPa] | $\mathrm{\varepsilon_{N}~{}[eV]}$ | $\mathrm{m_{h}}$ | $\mathrm{\alpha}$ | B[GPa] | $\mathrm{E_{g}^{KS}}$ | $\mathrm{E_{g}^{GW}}$
| $\mathrm{N=26}$ | 0.42 | -0.29 | -2.41 |  | 0.68 | -0.25 | -1.47 |  |  |
| $\mathrm{N=27}$ | 0.42 | -0.28 | -2.41 |  | 0.68 | -0.24 | -1.47 |  |  |
| $\mathrm{N=28}$ | 1.56 | -0.64 | -0.64 |  | 1.96 | -0.62 | -0.51 |  |  |
| $\mathrm{N=29}$ | 1.59 |  -0.65${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cap}$ | -0.63 |  | 1.97 | -0.63 | -0.51 |  |  |
$\mathrm{MoS_{2}-MoS_{2}}$ | $\mathrm{N=30}$ | 2.02 |  0.28${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cup}$ | -0.50 | 20.99 | 2.41 | 0.28 | -0.42 | 38.62 | 0.91 | 2.90
| $\mathrm{N=31}$ | 2.05 | 0.28 | -0.49 |  | 2.43 | 0.28 | -0.41 |  |  |
| $\mathrm{N=32}$ | 3.37 | 0.41 | -0.30 |  | 3.88 | 0.39 | -0.26 |  |  |
| $\mathrm{N=33}$ | 3.41 | 0.44 | -0.29 |  | 3.90 | 0.39 | -0.26 |  |  |
| $\mathrm{N=26}$ | 0.30 | -0.26 | -3.34 |  | 0.49 | -0.47 | -2.03 |  |  |
| $\mathrm{N=27}$ | 0.56 | -0.35 | -1.80 |  | 0.75 | -0.64 | -1.34 |  |  |
| $\mathrm{N=28}$ | 1.48 | -0.59 | -0.67 |  | 1.76 | -1.22 | -0.57 |  |  |
| $\mathrm{N=29}$ | 1.68 |  -0.77${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cap}$ | -0.60 |  | 2.01 | -1.62 | -0.50 |  |  |
$\mathrm{MoSeS-MoS_{2}}$ | $\mathrm{N=30}$ | 1.99 |  0.26${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cup}$ | -0.50 | 21.65 | 2.33 | 0.62 | -0.43 | 37.26 | 0.85 | 2.78
| $\mathrm{N=31}$ | 2.04 | 0.29 | -0.49 |  | 2.33 | 0.62 | -0.43 |  |  |
| $\mathrm{N=32}$ | 3.27 | 0.48 | -0.31 |  | 3.69 | 0.95 | -0.27 |  |  |
| $\mathrm{N=33}$ | 3.40 | 0.39 | -0.29 |  | 3.80 | 0.36 | -0.26 |  |  |
| $\mathrm{N=26}$ | 0.36 | -0.24 | -2.81 |  | 0.23 | -0.21 | -4.43 |  |  |
| $\mathrm{N=27}$ | 1.36 | -0.40 | -0.74 |  | 1.16 | -0.35 | -0.86 |  |  |
| $\mathrm{N=28}$ | 1.92 | -0.94 | -0.52 |  | 1.91 | -0.49 | -0.53 |  |  |
| $\mathrm{N=29}$ | 2.06 |  -0.49${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cap}$ | -0.49 |  | 1.95 | -0.88 | -0.51 |  |  |
$\mathrm{MoSe_{2}-MoS_{2}}$ | $\mathrm{N=30}$ | 2.10 |  0.25${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cup}$ | -0.48 | 20.21 | 2.11 | 0.24 | -0.48 | 36.08 | 0.06 | 2.14
| $\mathrm{N=31}$ | 2.70 | 0.28 | -0.37 |  | 2.58 | 0.29 | -0.39 |  |  |
| $\mathrm{N=32}$ | 3.30 | 0.57 | -0.30 |  | 3.39 | 0.54 | -0.30 |  |  |
| $\mathrm{N=33}$ | 4.08 | 0.38 | -0.25 |  | 4.06 | 0.37 | -0.25 |  |  |
| $\mathrm{N=26}$ | 0.59 | -0.32 | -1.69 |  | 0.88 | -0.27 | -1.13 |  |  |
| $\mathrm{N=27}$ | 0.60 | -0.30 | -1.68 |  | 0.88 | -0.27 | -1.13 |  |  |
| $\mathrm{N=28}$ | 1.61 | -0.58 | -0.62 |  | 2.01 | -0.57 | -0.50 |  |  |
| $\mathrm{N=29}$ | 1.66 |  -0.59${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cap}$ | -0.60 |  | 2.04 | -0.58 | -0.49 |  |  |
$\mathrm{MoSe_{2}-MoSe_{2}}$ | $\mathrm{N=30}$ | 2.01 |  0.24${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cup}$ | -0.50 | 17.17 | 2.41 | 0.24 | -0.42 | 33.43 | 0.27 | 1.95
| $\mathrm{N=31}$ | 2.07 | 0.24 | -0.48 |  | 2.45 | 0.24 | -0.41 |  |  |
| $\mathrm{N=32}$ | 3.27 | 0.44 | -0.31 |  | 3.77 | 0.44 | -0.27 |  |  |
| $\mathrm{N=33}$ | 3.36 | 0.49 | -0.30 |  | 3.83 | 0.46 | -0.26 |  |  |
| $\mathrm{N=26}$ | 0.34 | -0.60 | -2.96 |  | 0.77 | -0.24 | -1.31 |  |  |
| $\mathrm{N=27}$ | 0.59 | -0.74 | -1.71 |  | 1.02 | -0.31 | -0.98 |  |  |
| $\mathrm{N=28}$ | 1.43 | -1.10 | -0.70 |  | 1.94 | -0.50 | -0.52 |  |  |
| $\mathrm{N=29}$ | 1.61 |  -1.93${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cap}$ | -0.62 |  | 2.17 | -0.95 | -0.46 |  |  |
$\mathrm{MoSSe-MoSe_{2}}$ | $\mathrm{N=30}$ | 1.93 |  0.57${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}~{}\cup}$ | -0.52 | 18.50 | 2.47 | 0.27 | -0.40 | 34.63 | 0.64 | 2.71
| $\mathrm{N=31}$ | 1.99 | 0.54 | -0.50 |  | 2.50 | 0.26 | -0.40 |  |  |
| $\mathrm{N=32}$ | 3.16 | 0.98 | -0.32 |  | 3.81 | 0.44 | -0.26 |  |  |
| $\mathrm{N=33}$ | 3.31 | 0.89 | -0.30 |  | 3.93 | 0.40 | -0.26 |  |  |

Table 3: Calculations of effective mass of hole of parabolic constant,
$\mathrm{\alpha~{}[1/eV]}$ to corresponding energy of band index,
$\mathrm{\varepsilon_{N}}$ $\&$ Bulk modulus, B for $\mathrm{MoSSe-MoSeS}$ ,
$\mathrm{MoSeS-MoSSe}$ , $\&$ $\mathrm{MoSSe-MoSSe}$ with vdW-DF2 and GLLB-SC
exchange correlations plus PBE within GW approximation

|  |  | $\mathrm{\underline{vdW-DF2}}$ |  |  |  | $\mathrm{\underline{GLLB-SC}}$ |  |  |  | $\underline{\mathrm{G_{0}W_{0}@PBE}}$
---|---|---|---|---|---|---|---|---|---|---|---
System | $\mathrm{Band}$ | $\mathrm{\varepsilon_{N}~{}[eV]}$ | $\mathrm{m_{h}}$ | $\mathrm{~{}\alpha}$ | B[GPa] | $\mathrm{\varepsilon_{N}~{}[eV]}$ | $\mathrm{m_{h}}$ | $\mathrm{\alpha}$ | B[GPa] | $\mathrm{E_{g}^{KS}}$ | $\mathrm{E_{g}^{GW}}$
| $\mathrm{N=26}$ | 0.36 | -0.65 | -2.79 |  | 0.22 | -0.27 | -4.47 |  |  |
| $\mathrm{N=27}$ | 0.36 | -0.63 | -2.77 |  | 0.23 | -0.26 | -4.44 |  |  |
| $\mathrm{N=28}$ | 1.47 | -1.61 | -0.68 |  | 1.45 | -0.75 | -0.69 |  |  |
| $\mathrm{N=29}$ | 1.52 |  -1.49${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cap}$ | -0.66 |  | 1.48 | -0.71 | -0.67 |  |  |
$\mathrm{MoSSe-MoSeS}$ | $\mathrm{N=30}$ | 1.88 |  0.60${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cup}$ | -0.53 | 20.14 | 1.87 | 0.29 | -0.54 | 35.97 | 0.87 | 2.78
| $\mathrm{N=31}$ | 1.94 | 0.57 | -0.52 |  | 1.91 | 0.27 | -0.53 |  |  |
| $\mathrm{N=32}$ | 3.15 | 0.85 | -0.32 |  | 3.25 | 0.39 | -0.31 |  |  |
| $\mathrm{N=33}$ | 3.24 | 0.98 | -0.31 |  | 3.30 | 0.42 | -0.30 |  |  |
| $\mathrm{N=26}$ | 0.33 | -0.33 | -3.03 |  | 0.88 | -0.26 | -1.14 |  |  |
| $\mathrm{N=27}$ | 0.33 | -0.30 | -3.03 |  | 0.88 | -0.26 | -1.14 |  |  |
| $\mathrm{N=28}$ | 1.46 | -0.71 | -0.69 |  | 2.12 | -0.73 | -0.47 |  |  |
| $\mathrm{N=29}$ | 1.47 |  -0.73${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cap}$ | -0.68 |  | 2.12 | -0.73 | -0.47 |  |  |
$\mathrm{MoSeS-MoSSe}$ | $\mathrm{N=30}$ | 1.88 |  0.27${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cup}$ | -0.53 | 20.23 | 2.54 | 0.28 | -0.39 | 35.96 | 0.92 | 2.80
| $\mathrm{N=31}$ | 1.89 |  -0.28${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cap}$ | -0.53 |  | 2.54 | 0.28 | -0.39 |  |  |
| $\mathrm{N=32}$ | 3.16 |  0.43${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cup}$ | -0.32 |  | 3.93 | 0.40 | -0.26 |  |  |
| $\mathrm{N=33}$ | 3.17 | 0.44 | -0.32 |  | 3.94 | 0.40 | -0.25 |  |  |
| $\mathrm{N=26}$ | 0.42 | -0.30 | -2.38 |  | 0.39 | -0.26 | -2.56 |  |  |
| $\mathrm{N=27}$ | 0.91 | -0.30 | -1.10 |  | 0.79 | -0.26 | -1.27 |  |  |
| $\mathrm{N=28}$ | 1.55 |  -0.72${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cap}$ | -0.64 |  | 1.63 | -0.72 | -0.62 |  |  |
| $\mathrm{N=29}$ | 1.97 |  0.27${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cup}$ | -0.51 |  | 2.03 | -0.74 | -0.49 |  |  |
$\mathrm{MoSSe-MoSSe}$ | $\mathrm{N=30}$ | 2.04 |  -0.71${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cap}$ | -0.49 | 20.20 | 2.04 | 0.28 | -0.49 | 35.99 | 0.25 | 2.29
| $\mathrm{N=31}$ | 2.46 |  0.27${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}~{}\cup}$ | -0.41 |  | 2.45 | 0.28 | -0.41 |  |  |
| $\mathrm{N=32}$ | 3.26 | 0.43 | -0.31 |  | 3.43 | 0.41 | -0.29 |  |  |
| $\mathrm{N=33}$ | 3.75 | 0.43 | -0.27 |  | 3.84 | 0.40 | -0.26 |  |  |

Higher temperatures enhance the likelihood of discovering energetic electrons
and holes, hence the generation rate rises with temperature. For small
electron-hole densities, increasing temperature spreads carrier distributions
to higher energies where Auger recombination [33] is less efficient, reducing
the recombination rate. Large electron-hole concentrations require vacant
final scattering states for electron-hole recombination near the band
boundaries. An increase in temperature results in more vacant states. As a
result of the two variables mentioned above, recombination rates at high
electron-hole densities are less temperature sensitive. Thus, improving the
seebeck coefficient, either by designing the density of states or altering
nanostructure, is a requirement for producing effective thermoelectric
materials, and our findings support this argument in Refs [34, 35, 36].

Eq. 3 could be rearranged to form a parabolic band model for analyzing the
band alignment effect at interfaces as,

$\mathrm{E(k)=\frac{\hbar^{2}k^{2}}{2m^{*}}-\alpha E(k)^{2}}$ (5)

Where $\alpha$ is used to quantify the flattening of a parabola curve, a
smaller negative value indicates that the parabolic curve is more flat. Thus,
accurately estimating the effective mass from the $\mathrm{E(k)}$ curve is
crucial for correctly modeling the optical and transport properties of various
structural configurations. The Taylor expansion can be used to discover the
$\mathrm{E(k)}$ dispersion relation as follows:

$\mathrm{E(k)=E_{0}+\frac{\partial E}{\partial
k}+\frac{\partial^{2}E}{2!\partial k^{2}}+O(k^{3})}$ (6)

Where $\mathrm{E_{0}}$ is is energy at $\Gamma$-point. Then, contracting Eq.6
and discarding the second term, we get,

$\mathrm{E(k)=E_{0}+\alpha|\Delta k|^{2}}$ (7)

Thus, the change in $\mathrm{E(k)}$ curve at a symmetric path k-point divided
by the change in square of the k-point path equals twice the band curvature.

$\mathrm{\frac{\hbar^{2}}{2m^{*}}=\frac{\partial^{2}E}{2!\partial
k^{2}}=\alpha}$ (8)

$\mathrm{\frac{\hbar^{2}k^{2}}{2m^{*}}=\alpha}$, this relation proves that
materials with small masses have high mobility and long diffusion lengths
[37].

In this work, we investigated energy change, E(k), along the k-point path from
$\mathrm{\Gamma}$ to $\mathrm{K}$, where $\mathrm{G~{}[0.0,0.0,0.0]}$ and
$\mathrm{K~{}[1/2,1/2,0.0]}$ were used to calculate effective mass
($\mathrm{m_{e}~{}\&~{}m_{h}}$) and parabolic constant, $\mathrm{\alpha}$
while taking into account the transiting electron path of choice.

Table.1 illustrates, the computed effective mass, $\mathrm{m_{e}}$ of bilayer
$\mathrm{2H-MoS_{2}}$ and $\mathrm{2H-MoSe_{2}}$ is $\mathrm{0.69m_{e}}$ and
$\mathrm{0.75m_{e}}$, respectively, which are consistent with the literature
values of $\mathrm{0.7m_{e}}$ and $\mathrm{0.8m_{e}}$ given in Refs [38, 39,
40]. Van der Waals interactions between layers have a significant impact on a
material’s effective mass and band curvature. In this work, we analyzed
effective mass using the GLLBSC exchange correlation, and discovered that the
value is underestimated since the GLLBSC correlation ignores the London force.
Therefore, taking into account improved van der Waals interaction exchange
correlation yields the best results.

Table. 2 shows that electrons may excite either directly or indirectly
depending on band orientation. Excitation occurs when the effective hole mass
density changes sign, suggesting a leap that requires a particular amount of
energy, implying that band alignment has a substantial influence on exiting
energy. Consequently, direct transition excitation requires less energy, but
indirect transition excitation requires more exciting energy. Thus,
$\mathrm{MoSeS-MoS_{2}}$ and $\mathrm{MoSe_{2}-MoSe_{2}}$ have a relatively
smaller band confinement effect and hence require less energy due to direct
band alignment.

Table. 3 shows that the proportionate size of constituent atoms, when arranged
in various configurations, can result in varied band alignments effect. The
interaction of adjacent layers of the same type can enhance optical properties
such as multiple excitation peaks. Therefore, layer contact of various sorts
with adjacent sheets has the potential to improve the optical properties of
hetrostructure. One can see that both $\mathrm{MoSeS-MoSSe}$ and
$\mathrm{MoSSe-MoSSe}$ have double excitation peaks.

##  4 Conclusion

The tuning of 2D structures in various configurations is critical to optical
and thermoelectric properties. In this paper, we used the kane dispersion,
$\mathrm{E(k)}$ relation to calculate the density of hot electrons with their
corresponding group velocity, as well as the electronic gap,
$\mathrm{E_{g}^{KS}}$ which is the difference between the conduction band edge
energy, $\mathrm{~{}E_{c}=\frac{1}{\alpha_{c}}}$ and the valence band edge
energy, $\mathrm{~{}E_{v}=\frac{1}{\alpha_{v}}}$, which allowed us to
accurately determine $\mathrm{E_{g}^{opt}+E_{B}=E_{g}^{GW}-E_{g}^{KS}\approx
E_{g}^{BSE}}$. We also discovered how different configurations alter the
exciton binding energy and thermoelectric properties of a 2D material. In all
respects, the highest hetro configuration offers excellent thermoelectric
properties. Finally, we estimated bulk modulus, which is a measure of material
stability, using the Murnaghan curve fitting method, and we discovered that
hetrostructures stabilize themselves when layer interaction increases owing to
surface exposure to adjacent layers.

## Disclosure statement

The author declare that there is no conflict of interest.

##  5 Data Availability Statement

The data that support the findings of this study are available upon reasonable
request from the author.

##  6 Acknowledgements

We are grateful to the Dilla University for financial support.

ORCID iDs. T.E. Ada.
https://orcid.org/0000-0002-4417-0058.

##  7 References

## References

  * [1] S. Latini, K. T. Winther, T. Olsen, K. S. Thygesen, Interlayer excitons and band alignment in mos2/hbn/wse2 van der waals heterostructures., Nano letters 17.
URL https://api.semanticscholar.org/CorpusID:206737138

  * [2] X. Blase, I. Duchemin, D. Jacquemin, P.-F. Loos, The bethe–salpeter equation formalism: From physics to chemistry, The Journal of Physical Chemistry Letters 11.
URL https://doi.org/10.1021/acs.jpclett.0c01875

  * [3] F. A. Rasmussen, P. S. Schmidt, K. T. Winther, K. S. Thygesen, Efficient many-body calculations for two-dimensional materials using exact limits for the screened potential: Band gaps of ${\mathbf{mos}}_{2},h$-bn, and phosphorene, Phys. Rev. B 94.
URL https://link.aps.org/doi/10.1103/PhysRevB.94.155406

  * [4] B. Ridley, The diffusion of hot electrons across a semiconductor base, Solid-State Electronics 24.
URL https://doi.org/10.1016/0038-1101(81)90010-1

  * [5] S. Latini, T. Olsen, K. S. Thygesen, Excitons in van der waals heterostructures: The important role of dielectric screening, Phys. Rev. B 92.
URL https://link.aps.org/doi/10.1103/PhysRevB.92.245123

  * [6] J. Enkovaara, C. Rostgaard, J. Mortensen, J. Chen, M. Dułak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H. Hansen, H. Kristoffersen, M. Kuisma, A. Larsen, L. Lehtovaara, M. Ljungberg, O. Lopez-Acevedo, P. Moses, J. Ojanen, T. Olsen, V. Petzold, N. Romero, J. Stausholm-M${\rm\o}$ller, M. Strange, G. Tritsaris, M. Vanin, M. Walter, B. Hammer, H. H$\rm\ddot{a}$kkinen, G. Madsen, R. Nieminen, J. N${\rm\o}$rskov, M. Puska, T. Rantala, J. Schi${\rm\o}$tz, K. Thygesen, K. Jacobsen, Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method, J. Phys.: Condens. Matter 22 (2010) 253202.
URL https://doi.org/10.1088/0953-8984/22/25/253202

  * [7] J. Mortensen, L. Hansen, K. Jacobsen, Real-space grid implementation of the projector augmented wave method, Phys. Rev. B 71 (2005) 035109.
URL https://doi.org/10.1103/PhysRevB.71.035109

  * [8] W. Kohn, L. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) A1133–A1138.
URL https://doi.org/10.1103/PhysRev.140.A1133

  * [9] P. Bl$\rm\ddot{o}$chl, Projector augmented-wave method, Phys. Rev. B 50 (1994) 17953.
URL https://doi.org/10.1103/PhysRevB.50.17953

  * [10] H. Monkhorst, J. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188.
URL https://doi.org/10.1103/PhysRevB.13.5188

  * [11] G. Kresse, D. Joubert, From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method, Phys. Rev. B 59 (1999) 1758.
URL http://dx.doi.org/10.1103/PhysRevB.59.1758

  * [12] H. Schlegel, Optimization of equilibrium geometries and transition structures, J. Comp. Chem. 3 (1982) 214\.
URL https://doi.org/10.1002/jcc.540030212

  * [13] P. Feynman, Forces in Molecules, Phys. Rev. 56 (1939) 340.
URL https://doi.org/10.1103/PhysRev.56.340

  * [14] O. Nielsen, R. Martin, Quantum-mechanical theory of stress and force, Phys. Rev. B. 32 (1985) 3780.
URL https://doi.org/10.1103/PhysRevB.32.3780

  * [15] R. Wentzcovitch, J. Martins, First principles molecular dynamics of Li: Test of a new algorithm, Solid State Commun. 78 (1991) 831.
URL https://doi.org/10.1016/0038-1098(91)90629-A

  * [16] J. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77 (1996) 3865.
URL https://doi.org/10.1103/PhysRevLett.77.3865

  * [17] M. Kuisma, J. Ojanen, J. Enkovaara, T. Rantala, Kohn-Sham potential with discontinuity for band gap materials, Phys. Rev. B 82 (2010) 115106.
URL https://doi.org/10.1103/PhysRevB.82.115106

  * [18] J. Perdew, A. Ruzsinszky, G. Csonka, O. Vydrov, G. Scuseria, L. Constantin, X. Zhou, K. Burke, Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces, Erratum Phys. Rev. Lett. 102 (2009) 039902.
URL https://doi.org/10.1103/PhysRevLett.100.136406

  * [19] G. N. Koskowich, M. Soma, R. B. Darling, Near-infrared free-carrier optical absorption in silicon: Effect of first-order phonon-assisted scattering in a nonparabolic conduction band, Phys. Rev. B 41\.
URL https://link.aps.org/doi/10.1103/PhysRevB.41.2944

  * [20] B. Hammer, L. B. Hansen, J. K. Nørskov, Improved adsorption energetics within density-functional theory using revised perdew-burke-ernzerhof functionals, Phys. Rev. B 59.
URL https://link.aps.org/doi/10.1103/PhysRevB.59.7413

  * [21] J. Klimes, D. R. Bowler, A. Michaelides, Chemical accuracy for the van der waals density functional, Journal of Physics: Condensed Matter 22 (2) (2009) 022201.
URL https://dx.doi.org/10.1088/0953-8984/22/2/022201

  * [22] D. Chakraborty, K. Berland, T. Thonhauser, Next-generation nonlocal van der waals density functional, Journal of Chemical Theory and Computation 16.
URL https://doi.org/10.1021/acs.jctc.0c00471

  * [23] B. R. Nag, A. N. Chakravarti, On a simplified form of kane’s dispersion relation for semiconductors, physica status solidi (b) 71.
URL https://doi.org/10.1002/pssb.2220710153

  * [24] J. Kang, S. Tongay, J. Zhou, J. Li, J. Wu, Band offsets and heterostructures of two-dimensional semiconductors, Applied Physics Letters 102.
URL https://doi.org/10.1063/1.4774090

  * [25] F. Huser, T. Olsen, K. S. Thygesen, Quasiparticle gw calculations for solids, molecules, and two-dimensional materials, Phys. Rev. B 87.
URL https://link.aps.org/doi/10.1103/PhysRevB.87.235132

  * [26] V. Jindal, D. Jana, T. Deilmann, S. Ghosh, Interlayer and excited-state exciton transitions in bulk $2h\text{$-$}{\mathrm{mos}}_{2}$, Phys. Rev. B 102.
URL https://link.aps.org/doi/10.1103/PhysRevB.102.235204

  * [27] M. Bhatnagar, T. Woźniak, Ł. Kipczak, N. Zawadzka, K. Olkowska-Pucko, M. Grzeszczyk, J. Pawłowski, K. Watanabe, T. Taniguchi, A. Babiński, M. R. Molas, Temperature induced modulation of resonant raman scattering in bilayer 2h-mos2, Scientific Reports 12.
URL https://doi.org/10.1038/s41598-022-18439-7

  * [28] P. Tonndorf, R. Schmidt, P. Böttger, X. Zhang, J. Börner, A. Liebig, M. Albrecht, C. Kloc, O. Gordan, D. R. T. Zahn, S. M. de Vasconcellos, R. Bratschitsch, Photoluminescence emission and raman response of monolayer mos2, mose2, and wse2, Opt. Express 21\.
URL https://opg.optica.org/oe/abstract.cfm?URI=oe-21-4-4908

  * [29] H. J. Liu, L. Jiao, L. Xie, F. Yang, J. L. Chen, W. K. Ho, C. L. Gao, J. F. Jia, X. D. Cui, M. H. Xie, Molecular-beam epitaxy of monolayer and bilayer wse2: a scanning tunneling microscopy/spectroscopy study and deduction of exciton binding energy, 2D Materials 2.
URL https://dx.doi.org/10.1088/2053-1583/2/3/034004

  * [30] T. Olsen, S. Latini, F. Rasmussen, K. S. Thygesen, Simple screened hydrogen model of excitons in two-dimensional materials, Phys. Rev. Lett. 116.
URL https://link.aps.org/doi/10.1103/PhysRevLett.116.056401

  * [31] R. Aksoy, Y. Ma, E. Selvi, M. C. Chyu, A. Ertas, A. White, X-ray diffraction study of molybdenum disulfide to 38.8gpa, Journal of Physics and Chemistry of Solids 67.
URL https://www.sciencedirect.com/science/article/pii/S0022369706002769

  * [32] M. Hebbache, M. Zemzemi, Ab initio study of high-pressure behavior of a low compressibility metal and a hard material: Osmium and diamond, Phys. Rev. B 70.
URL https://link.aps.org/doi/10.1103/PhysRevB.70.224107

  * [33] Y. Jiang, M. Cui, S. Li, C. Sun, Y. Huang, J. Wei, L. Zhang, M. Lv, C. Qin, Y. Liu, M. Yuan, Reducing the impact of auger recombination in quasi-2d perovskite light-emitting diodes, Nature Communications 12.
URL https://doi.org/10.1038/s41467-020-20555-9

  * [34] Y. Pei, A. D. LaLonde, H. Wang, G. J. Snyder, Low effective mass leading to high thermoelectric performance, Energy Environ. Sci. 5.
URL http://dx.doi.org/10.1039/C2EE21536E

  * [35] D. K. Ferry, First-order optical and intervalley scattering in semiconductors, Phys. Rev. B 14.
URL https://link.aps.org/doi/10.1103/PhysRevB.14.1605

  * [36] F. Rana, Electron-hole generation and recombination rates for coulomb scattering in graphene, Phys. Rev. B 76.
URL https://link.aps.org/doi/10.1103/PhysRevB.76.155431

  * [37] F. Brivio, K. T. Butler, A. Walsh, M. van Schilfgaarde, Relativistic quasiparticle self-consistent electronic structure of hybrid halide perovskite photovoltaic absorbers, Phys. Rev. B 89.
URL https://link.aps.org/doi/10.1103/PhysRevB.89.155204

  * [38] T. Cheiwchanchamnangij, W. R. L. Lambrecht, Quasiparticle band structure calculation of monolayer, bilayer, and bulk $\mathrm{MoS_{2}}$, Phys. Rev. B 85.
URL https://link.aps.org/doi/10.1103/PhysRevB.85.205302

  * [39] R. Pisoni, T. Davatz, K. Watanabe, T. Taniguchi, T. Ihn, K. Ensslin, Absence of interlayer tunnel coupling of $k$-valley electrons in bilayer ${\mathrm{mos}}_{2}$, Phys. Rev. Lett. 123 (2019) 117702.  doi:10.1103/PhysRevLett.123.117702.
URL https://link.aps.org/doi/10.1103/PhysRevLett.123.117702

  * [40] S. Larentis, H. C. P. Movva, B. Fallahazad, K. Kim, A. Behroozi, T. Taniguchi, K. Watanabe, S. K. Banerjee, E. Tutuc, Large effective mass and interaction-enhanced zeeman splitting of k-valley electrons in $\mathrm{MoSe_{2}}$, Phys. Rev. B 97.
URL https://link.aps.org/doi/10.1103/PhysRevB.97.201407