# Effects of Co-channel Interference on RIS Empowered Wireless Networks amid
Multiple Eavesdropping Attempts

Md. Roisul Ajom Ruku  Department of Electrical & Electronic Engineering,
Rajshahi University of Engineering & Technology (RUET), Rajshahi-6204,
Bangladesh  Md. Ibrahim  Institute of Information and Communication
Technology, RUET  A. S. M. Badrudduza  Department of Electronics &
Telecommunication Engineering,RUET  Imran Shafique Ansari  James Watt School
of Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom

###### Abstract

## Abstract

This letter is concerned with the secrecy performance of reconfigurable
intelligent surfaces (RIS)-aided wireless networks in the existence of
multiple interferers towards the destination. To be more precise, we analyze
three critical issues in the design of secure RIS-assisted networks: 1) How do
interferers affect the performance of secure wireless networks? 2) Which of
the two groups of eavesdroppers (i.e., colluding and non-colluding) is more
severe? 3) How can RIS improve network confidentiality? To do so, we develop
the analytical expression of secrecy outage probability in closed-form, along
with asymptotic analysis at high signal-to-noise ratio regime to better
understand the impacts of different system parameters on secrecy performance.
Finally, we validate our analytical results using a computer based Monte-Carlo
simulation.

## Keywords

Reconfigurable intelligent surface (RIS), co-channel interference (CCI),
secrecy outage probability (SOP), eavesdropper

##  1 Introduction

###  1.1 Background and Literature Study

Reconfigurable intelligent surfaces (RIS) have become a popular topic in
sixth-generation ($6$G) wireless networks due to environmental and economic
considerations, as they can increase coverage, enhance energy efficiency, and
reduce costs [1]. In general, RIS consists of numerous passive elements that
form a flat meta surface, and these components can be controlled to optimize
the phase and amplitude of incoming signals, thereby reducing negative effects
on wireless networks [2]. Unlike other technologies, the signal does not
undergo encoding and decoding during transmission, which has generated
interest in RIS as a field of research.

In light of the broadcast nature of the wireless channel, information security
has emerged as one of the most pressing issues in next-generation
communication. However, physical layer security (PLS) analysis, which ensures
confidential transmission perfectly, has been proven to be an effective
solution in this context [3]. Despite being a new technology, the PLS of RIS-
aided models has been studied in a few articles [4, 5, 6, 7, 8] only. For
example, the authors of [4] suggested that deploying RIS between source and
destination could improve secrecy performance significantly. In [5], the
authors studied the PLS for RIS-aided networks with multiple eavesdroppers and
demonstrated how antenna diversity can facilitate secrecy performance.
Moreover, integrating RIS in non-orthogonal multiple access networks, secrecy
behavior was evaluated recently in [7, 8]. Besides, co-channel interference
(CCI), particularly in cellular communications, is inevitable due to the
spatial reuse of the frequency spectrum in wireless networks [9]. Two relay-
assisted multi-user scheduling techniques were proposed in [10] for
investigating secrecy characteristics in the presence of CCI. A research in
[11] reported recently that increasing the number of interferers towards the
destination drastically affects secrecy performance.

###  1.2 Motivation and Contributions

According to the aforementioned studies, the existing literature is generally
focused on RIS-aided configurations where Nakagami-$m$ distributions are
largely taken into account. Moreover, no research has been conducted yet to
realize the impact of CCI on the secrecy performance of such model. In
contrast to the current literature, we propose a novel RIS-assisted structure
under $\alpha$-$\mu$ distributions wherein multiple interferers affect the
destination and eavesdroppers are capable of intercepting information directly
from the source or via RIS. We also assume two eavesdropping cases i.e.
colluding and non-colluding (NC) schemes in our analysis. Therefore, taking
into account generalized $\alpha$-$\mu$ fading models provides the benefits of
incorporating some previous research [4, 6] as part of our work. To summarize,
the key contributions of this paper are given below:

  * •

Unlike previous works, which mainly focus on Nakagami-$m$ and Rayleigh
distributions for RIS-assisted cellular networks, we introduce $\alpha$-$\mu$
fading models in our analysis for the first time. Moreover, we investigate a
general PLS framework for such configuration in the presence of CCIs that is
novel to the author’s best knowledge.

  * •

To analyze the secrecy performance, firstly, we derive the cumulative density
function (CDF) for the RIS-aided network. Therefore, we develop novel
analytical expressions of secrecy outage probability (SOP) applicable in three
different scenarios. More precisely, asymptotic secrecy analysis is also
provided for the considered scenarios including colluding and NC eavesdropping
attacks to provide some insightful information.

  * •

Finally, based on these demonstrated expressions, we conclude the discussion
by providing some insight into the design of secure RIS-aided wireless
networks. Furthermore, the analytical results are also verified with the aid
of Monte-Carlo (MC) simulations.

###  1.3 Organization

The outline of this paper is designed as follows: Section 2 delineates models
of the proposed system and channel in realization, while Section 3
demonstrates the analytical expressions of SOP for two different
eavesdroppers, as well as their asymptotic expressions. Numerical outcomes and
some important observations are provided in Section 4. Finally, the paper
concludes with Section 5.

##  2 System Model and Problem Formulation

In this paper, a RIS-assisted wireless network is considered where the source
($\mathcal{S}$) aims to communicate sensitive information to the receiver
($\mathcal{D}$) with the help of a RIS ($\mathcal{R}$) that is furnished with
$\mathcal{N}_{d}$ passive elements while multiple colluding (Case-I) and non-
colluding (Case-II) eavesdroppers ($\mathcal{E}$) attempt to intercept the
confidential communication directly (Scenario-I) or via another RIS
($\mathcal{I}$) (Scenario-II) having $\mathcal{N}_{E}$ reflecting elements or
via RIS ($\mathcal{R}$) (Scenario-III). We assume $\mathcal{D}$ is affected by
the interfering signals from $\mathcal{M}$ arbitrary number of interferers.
Here, each node (e.g. $\mathcal{S}$ and $\mathcal{D}$) is comprised of a
single antenna, and all the links e.g. $\mathcal{S}$-$\mathcal{R}_{k}$
($k=1,2,\ldots,\mathcal{N}_{d}$), $\mathcal{R}_{k}$-$\mathcal{D}$,
$\mathcal{S}$-$\mathcal{E}_{q}$ ($q=1,2,\ldots,\mathcal{L}$),
$\mathcal{S}$-$\mathcal{I}_{p}$ ($p=1,2,\ldots,\mathcal{N}_{E}$),
$\mathcal{I}_{p}$-$\mathcal{E}_{q}$, and $\mathcal{M}_{i}$-$\mathcal{D}$
($i=1,2,\ldots,\mathcal{M}$) are subjected to $\alpha$-$\mu$ fading
distribution with corresponding channel gains of
$h_{sk}=\theta_{k}e^{-j\delta_{k}},h_{kd}=\phi_{k}e^{-j\beta_{k}},h_{sq},h_{sp}=\theta_{p}e^{-j\delta_{p}},h_{pq}=\phi_{pq}e^{-j\beta_{pq}},h_{id}$,
respectively, where $\theta_{k}$, $\phi_{k}$, $\theta_{p}$, $\phi_{pq}$ are
the channel amplitudes and $\delta_{k}$, $\beta_{k}$, $\delta_{p}$,
$\beta_{pq}$ refer to the phase amplitudes. Let $\mathcal{P}_{s}$ denotes the
transmit signal power from the source, $\mathcal{P}_{i}$ is the average power
of the transmit signal from $i$th interferer, and $\mathcal{Z}_{d}$ and
$\mathcal{Z}_{eq}$ are the noise powers at $\mathcal{D}$ and $q$th
$\mathcal{E}$, respectively. Then, the instantaneous signal-to-noise ratios
(SNRs) of $\mathcal{M}_{i}-\mathcal{D}$ and $\mathcal{S}-\mathcal{E}_{q}$
links are given by $\gamma_{id}=\bar{\gamma}_{id}|h_{id}|^{2}$,
$\gamma_{sq}=\bar{\gamma}_{sq}|h_{sq}|^{2}$,
$\gamma_{e,c}=\sum_{q=1}^{\mathcal{L}}\gamma_{sq}$ (Case-I), and
$\gamma_{e,nc}=\arg\underset{q=1,2,\ldots,\mathcal{L}}{max}\gamma_{sq}$ (Case-
II), where the average SNR terms are defined as
$\bar{\gamma}_{id}=\frac{\mathcal{P}_{i}}{\mathcal{Z}_{d}}$ and
$\bar{\gamma}_{sq}=\frac{\mathcal{P}_{s}}{\mathcal{Z}_{eq}}$. Instantaneous
SNRs at $\mathcal{D}$ and $\mathcal{E}$ with the RIS links are, respectively,
given by
$\gamma_{d}=\frac{\mathcal{P}_{s}}{\mathcal{Z}_{d}}\bigl{[}\sum_{k=0}^{\mathcal{N}_{d}}\theta_{k}\phi_{k}e^{-j(\Theta_{k}-\delta_{k}-\beta_{k})}\bigl{]}^{2}$
and
$\gamma_{eq}=\frac{\mathcal{P}_{s}}{\mathcal{Z}_{eq}}\bigl{[}\sum_{p=0}^{\mathcal{N}_{E}}\theta_{p}\phi_{pq}e^{-j(\Theta_{pq}-\delta_{p}-\beta_{pq})}\bigl{]}^{2}$.
In order to maximize the SNRs, the optimal values are chosen as
$\Theta_{k}=\delta_{k}+\beta_{k}$ and $\Theta_{pq}=\delta_{p}+\beta_{pq}$, and
accordingly the corresponding SNRs are given by
$\gamma_{d}=\bar{\gamma}_{d}\bigl{[}\sum_{k=0}^{\mathcal{N}_{d}}\theta_{k}\phi_{k}\bigl{]}^{2}$
and
$\gamma_{eq}=\bar{\gamma}_{eq}\bigl{[}\sum_{p=0}^{\mathcal{N}_{E}}\theta_{p}\phi_{p}\bigl{]}^{2}$,
and $\gamma_{E,C}=\sum_{q=1}^{\mathcal{L}}\gamma_{eq}$ (Case-I) and
$\gamma_{E,NC}=\arg\underset{q=1,2,\ldots,\mathcal{L}}{max}\gamma_{eq}$ (Case-
II), where $\bar{\gamma}_{d}=\frac{\mathcal{P}_{s}}{\mathcal{Z}_{d}}$ and
$\bar{\gamma}_{eq}=\frac{\mathcal{P}_{s}}{Z_{eq}}$ are the average SNRs. Since
$\mathcal{D}$ receives interference from $\mathcal{M}$ interferers, the SIR at
$\mathcal{D}$ is
$\gamma_{d,i}=\frac{\mathcal{P}_{s}\bigl{[}\sum_{k=0}^{\mathcal{N}_{d}}\theta_{k}\phi_{k}\bigl{]}^{2}}{\sum_{i=1}^{\mathcal{M}}\mathcal{P}_{i}|h_{id}|^{2}}$.

###  2.1 PDF and CDF of $\gamma_{d}$

The probability density function (PDF) and the CDF of $\gamma_{d}$ can be
expressed as

$\displaystyle f_{\gamma_{d}}(\gamma)$
$\displaystyle=\frac{(\frac{\beta_{d}}{\bar{\gamma_{d}}})^{\frac{\vartheta_{d}}{2}}}{2\Gamma{(\vartheta_{d})}\zeta_{d}^{\vartheta_{d}}}\gamma^{\frac{\vartheta_{d}}{2}-1}e^{-\sqrt{\frac{\beta_{d}}{\bar{\gamma_{d}}\zeta_{d}^{2}}\gamma}},$
(1) $\displaystyle F_{\gamma_{d}}(\gamma)=$ $\displaystyle
1-\sum_{\Lambda_{1}=0}^{\vartheta_{d}-1}\frac{1}{\Lambda_{1}!}\left(\sqrt{\frac{\beta_{d}}{\bar{\gamma_{d}}\zeta_{d}^{2}}}\right)^{\Lambda_{1}}\gamma^{\frac{\Lambda_{1}}{2}}e^{-\sqrt{\frac{\beta_{d}}{\bar{\gamma_{d}}\zeta_{d}^{2}}\gamma}},$
(2)

where $\beta_{d}$ defines the path loss of the RIS-aided path and
$\bar{\gamma_{d}}$ denotes the average SNR of
$\mathcal{S}-\mathcal{R}_{k}-\mathcal{D}$ link.

Proof: Let $Y_{k}=\theta_{k}\phi_{k}$ be the product of two $\alpha$-$\mu$
random variables (RVs) where $\theta_{k}$ denotes the
$\mathcal{S}$-$\mathcal{R}_{k}$ hop with $\alpha_{s}$ and $\beta_{s}$ being
the fading parameters, and $\Omega_{s}$ be the scale parameter whereas
$\phi_{k}$ defines the $\mathcal{R}_{k}$-$\mathcal{D}$ hop with $\alpha_{r}$
and $\beta_{r}$ being the fading parameters, and $\Omega_{r}$ be the scale
parameter. Therefore, the PDF of $Y_{k}$ can be expressed as

$\displaystyle
f_{Y_{k}}(y)=\chi_{1}y^{\chi_{4}}K_{\chi_{2}}(\chi_{5}y^{\frac{\alpha_{s}}{2}}),$
(3)

where
$\chi_{1}=\frac{2\alpha_{s}\alpha_{r}\mu_{s}^{\mu{s}}\mu_{r}^{\mu{r}}}{\alpha_{s}\Gamma{(\mu_{s})}\Gamma{(\mu_{r})}\Omega_{s}^{\alpha_{s}\mu_{s}}\Omega_{r}^{\alpha_{r}\mu_{r}}}(\frac{\mu_{r}\Omega{s}^{\alpha_{s}}}{\mu_{s}\Omega{r}^{\alpha_{r}}})^{\frac{\alpha_{s}\mu_{s}-\alpha_{r}\mu_{r}}{2\alpha}},\chi_{2}=\frac{\alpha_{s}\mu_{s}-\alpha_{r}\mu_{r}}{\alpha}$,
$\chi_{3}=\frac{\mu_{r}\mu_{s}}{\Omega{r}^{\alpha_{r}}\Omega{s}^{\alpha_{s}}}$,
$\chi_{4}=\frac{\alpha_{s}\mu_{s}+\alpha_{r}\mu_{r}}{2}-1$,
$\chi_{5}=2\sqrt{\chi_{3}}$, and $K_{\nu}(\cdot)$ denotes the modified
$\nu^{th}$-order Bessel function [12, Eq. (8.432)]. However, with the help of
[13, Sec. (2.2.2)], it can be stated that the PDF of
$\tilde{Y}=\sum_{k=1}^{N_{d}}Y_{k}$ is tightly approximated by the first term
of a Laguerre expansion. Hence, the corresponding PDF can be expressed as

$\displaystyle
f_{\tilde{Y}}(y)\simeq\frac{1}{\Gamma{(\vartheta_{d}})\zeta_{d}^{\vartheta_{d}}}y^{\vartheta_{d}-1}e^{-\frac{y}{\zeta_{d}}},$
(4)

where $\vartheta_{d}$ denotes the shape parameter and $\zeta_{d}$ defines the
scale parameter, respectively, and they are related to the $s^{th}$ moment of
$Y_{k}$, can be expressed as

$\displaystyle\mathbb{E}(Y_{k}^{s})=\chi_{1}\frac{2^{s+\chi_{4}-1}}{\chi_{5}^{(s+\chi_{4}+1)}}\Gamma(\frac{\chi_{6}+s}{2})\Gamma(\frac{\chi_{7}+s}{2}),$
(5)

where $\chi_{6}=1+\chi_{4}+\chi_{2}$, $\chi_{7}=1+\chi_{4}-\chi_{2}$, and
$\Gamma(\cdot)$ is the Gamma function as defined in [12, Eq. (8.310)]. Now,
utilizing the RVs transformation technique and performing some mathematical
manipulations, the PDF of $\gamma_{d}$ can be finally shown as (1). Therefore,
the CDF of $\gamma_{d}$ can be written mathematically as

$\displaystyle F_{\gamma_{d}}(\gamma)=\int_{0}^{\gamma}f_{\tilde{Y}}(y)dy.$
(6)

Substituting (1) into (6) and utilizing [12, Eq. (3.381.8)], the CDF can be
obtained. Thus, the proof is completed.

###  2.2 PDF and CDF of $\gamma_{d,i}$

Assuming an interference limited system, the PDF of $\gamma_{d,i}$ is
expressed as

$\displaystyle
f_{\gamma_{d,i}}(\gamma)=\Xi_{2}\gamma^{\frac{\vartheta_{d}}{2}-1}G_{\mathcal{B}_{1},2}^{2,\mathcal{B}_{1}}\left[\mathcal{A}_{1}\gamma\bigg{|}\begin{array}[]{c}\alpha_{i}-\omega_{0}-1\\\
0,\frac{1}{2}\\\ \end{array}\right],$ (9)

where
$\mathcal{A}_{1}$=$\frac{\Xi_{1}^{2}(\mathcal{B}_{1}\bar{\gamma_{i}}^{\alpha_{i}})^{\mathcal{B}_{1}}}{4(\mathcal{M}\mu_{i})^{\mathcal{B}_{1}}}$,
$\mathcal{B}_{1}=\frac{1}{\alpha_{i}}$,
$\omega_{0}=\frac{\vartheta_{d}}{2}+\mathcal{M}\alpha_{i}\mu_{i}-1$,
$\Xi_{2}=\frac{(\mathcal{M}\mu_{i})^{\mathcal{M}\mu_{i}-\mathcal{B}_{1}(\omega_{0}+1)}\mathcal{B}_{1}^{\mathcal{B}_{1}(\omega_{0}+1)+\frac{1}{2}}\bar{\gamma}_{i}^{\alpha_{i}\mathcal{B}_{1}(\omega_{0}+1)-\mathcal{M}\alpha_{i}\mu_{i}}\alpha_{i}\Xi_{1}}{\sqrt{\pi}\sqrt{2}^{\mathcal{B}_{1}-1}\Gamma{(\mathcal{M}\mu_{i})}}$,
$\alpha_{i}$ and $\mu_{i}$ are the fading parameters of
$\mathcal{M}_{i}-\mathcal{D}$ link, $\bar{\gamma_{i}}$ denotes the average SNR
of $\mathcal{M}_{i}-\mathcal{D}$ link and $G[\cdot]$ is the Meijer’s G
function defined in [3]. Accordingly, the CDF of $\gamma_{d,i}$ is expressed
as

$\displaystyle
F_{\gamma_{d,i}}(\gamma)=\Xi_{2}\gamma^{\frac{\vartheta_{d}}{2}}G_{\mathcal{B}_{1}+1,3}^{2,\mathcal{B}_{1}+1}\left[\mathcal{A}_{1}\gamma\bigg{|}\begin{array}[]{c}\frac{-\vartheta_{d}}{2}+1,-\omega_{0}\\\
0,\frac{1}{2},\frac{-\vartheta_{d}}{2}\end{array}\right].$ (12)

Proof: Assuming all the interferers with equal power case, the PDF of
$\mathcal{M}_{i}$-$\mathcal{D}$ link can be expressed as

$\displaystyle
f_{\gamma_{i}}(x)=\frac{\alpha_{i}(\mathcal{M}\mu_{i})^{\mathcal{M}\mu_{i}}}{\Gamma(\mathcal{M}\mu_{i})\bar{\gamma}_{i}^{\alpha_{i}\mathcal{M}\mu_{i}}}x^{\alpha_{i}\mathcal{M}\mu_{i-1}}e^{-\frac{\mathcal{M}\mu_{i}}{\bar{\gamma}_{i}^{\alpha_{i}}}x^{\alpha_{i}}}.$
(13)

Now, the PDF of $\gamma_{d,i}$ can be written as

$\displaystyle
f_{\gamma_{d,i}}(\gamma)=\int_{0}^{\infty}xf_{\gamma_{d}}(\gamma
x)f_{\gamma_{i}}(x)dx.$ (14)

Substituting (1) and (13) into (14), the PDF of $\gamma_{d,i}$ as in (9) is
obtained. Subsequently, replacing (9) into (6) and with the aid of [14, Eq.
(2.24.2.2)], $F_{\gamma_{d,i}}$ can be derived. This concludes the proof.

###  2.3 PDF of $\gamma_{e}$

Scenerio-I: Here, we consider all the $\mathcal{S}$-$\mathcal{E}$ link undergo
$\alpha$-$\mu$ fading distributions. Based on the eavesdropping
characteristic, colluding (Case-I) and non-colluding (Case-II) cases are
considered.
Case-I: In the colluding eavesdropper scheme, eavesdroppers collaborate in
order to intercept legitimate information. Hence, the PDF of $\gamma_{e,c}$
can be expressed as

$\displaystyle
f_{\gamma_{e,c}}(\gamma)=\Xi_{3}\gamma^{(\mathcal{L}\alpha_{e}\mu_{e}-1)}e^{-\Xi_{4}\gamma^{\alpha_{e}}},$
(15)

where $\alpha_{e}$ and $\mu_{e}$ denote the fading parameters of
$\mathcal{S}-\mathcal{E}_{q}$ links,
$\Xi_{3}=\frac{\alpha_{e}(\mathcal{L}\mu_{e})^{\mathcal{L}\mu_{e}}}{\Gamma(\mathcal{L}\mu_{e})\bar{\gamma_{e}}^{\alpha_{e}\mathcal{L}\mu_{e}}}$,
$\Xi_{4}=\frac{\mathcal{L}\mu_{e}}{\bar{\gamma_{e}}^{\alpha_{e}}}$, and
$\bar{\gamma_{e}}$ denotes the average SNR of each links.

Case-II: Typically, in a NC situation, each eavesdropper performs an
individual decoding process to salvage the legitimate signal. Therefore, the
PDF of $\gamma_{e,nc}$ can be expressed as

$\displaystyle f_{\gamma_{e,nc}}(\gamma)$
$\displaystyle=\sum_{\Lambda_{3}=0}^{\infty}\sum_{k_{1}+...+k_{\mu_{e}-1}=\Lambda_{3}}\Xi_{5}\gamma^{\frac{\alpha_{e}\mu_{e}}{2}-1+\Xi_{6}}$
$\displaystyle\times$ $\displaystyle
e^{-(\Lambda_{3}+1)\mu_{e}(\frac{\gamma}{\bar{\gamma_{e}}})^{\frac{\alpha_{e}}{2}}},$
(16)

where
$\Xi_{5}=\binom{\mathcal{L}-1}{\Lambda_{3}}\binom{\Lambda_{3}}{k_{1}...k_{\mu_{e}-1}}\frac{\alpha_{e}\mu_{e}^{\mu_{e}+\Lambda_{2}\Lambda_{3}}\mathcal{L}}{2\Gamma{\mu_{e}}(\Lambda_{2}!)^{\Lambda_{3}}(-1)^{\Lambda_{3}}}\bar{\gamma_{e}}^{-\frac{\alpha_{e}\Lambda_{2}\Lambda_{3}+\alpha_{e}\mu_{e}}{2}}$
and
$\Xi_{6}=\sum_{\Lambda_{2}}\frac{\alpha_{e}}{2}\Lambda_{2}k_{\Lambda_{2}}$.
Proof: The PDF for NC eavesdropper can be written mathematically as

$\displaystyle
f_{\gamma_{e,nc}}(\gamma)=\mathcal{L}\left[F_{\gamma_{e,c}}(\gamma)\right]^{\mathcal{L}-1}f_{\gamma_{e,c}}(\gamma).$
(17)

With the help of [15, Eqs. (4-5)] and [12, Eq. (1.110)], and then performing
some algebraic manipulations, we obtain

$\displaystyle f_{e,nc}=$
$\displaystyle\sum_{\Lambda_{3}=0}^{\infty}\binom{\mathcal{L}-1}{\Lambda_{3}}\frac{\mathcal{L}\alpha_{e}\mu_{e}^{\mu_{e}+\Lambda_{2}\Lambda_{3}}}{2\Gamma{\mu_{e}}(\Lambda_{2}!)^{\Lambda_{3}}(-1)^{\Lambda_{3}}}\bar{\gamma_{e}}^{-\frac{\alpha_{e}\Lambda_{2}\Lambda_{3}+\alpha_{e}\mu_{e}}{2}}$
$\displaystyle\times$
$\displaystyle\gamma^{\frac{\alpha_{e}\mu_{e}}{2}-1}e^{-(\Lambda_{3}+1)\mu_{e}(\frac{\gamma}{\bar{\gamma_{e}}})^{\frac{\alpha_{e}}{2}}}\left(\sum_{\Lambda_{2}=0}^{\mu_{e}-1}\gamma^{\frac{\alpha_{e}\Lambda_{2}}{2}}\right)^{\Lambda_{3}}.$
(18)

Now, the expansion of
$\left(\sum_{\Lambda_{2}=0}^{\mu_{e}-1}\gamma^{\frac{\alpha_{e}\Lambda_{2}}{2}}\right)^{\Lambda_{3}}$
is written as

$\displaystyle\left(\sum_{\Lambda_{2}=0}^{\mu_{e}-1}\gamma^{\frac{\alpha_{e}\Lambda_{2}}{2}}\right)^{\Lambda_{3}}=$
$\displaystyle\sum_{k_{1}+..+k_{\mu_{e}-1}=\Lambda_{3}}\binom{\Lambda_{3}}{k_{1},...,k_{\mu_{e}-1}}$
$\displaystyle\times$
$\displaystyle\gamma^{\frac{\alpha_{e}}{2}\sum_{\Lambda_{2}}\Lambda_{2}k_{\Lambda_{2}}}.$
(19)

Finally, substituting (2.3) into (2.3), the proof is completed.

Scenario-II: Here, through $\mathcal{I}_{p}$, the eavesdroppers gain access to
the information they need to overhear. It is important to note that
$\mathcal{D}$ and $\mathcal{E}$ use two different RISs, i.e.,
$\mathcal{R}_{k}$ and $\mathcal{I}_{p}$, respectively.

Case-I: According to (1) and (2), the PDF and CDF for colluding eavesdroppers
can be respectively expressed as

$\displaystyle f_{\gamma_{E,C}}(\gamma)$
$\displaystyle=\Xi_{7}\gamma^{\frac{\vartheta_{E}}{2}-1}e^{-\Xi_{8}\sqrt{\gamma}},$
(20) $\displaystyle F_{\gamma_{E,C}}(\gamma)$
$\displaystyle=1-\sum_{\Lambda_{4}=0}^{\vartheta_{E}-1}\frac{\Xi_{8}^{\Lambda_{4}}}{\Lambda_{4}!}\gamma^{\frac{\Lambda_{4}}{2}}e^{-\Xi_{8}\sqrt{\gamma}},$
(21)

where
$\Xi_{7}=(\frac{\beta_{E}}{\bar{\gamma}_{E}})^{\frac{\vartheta_{E}}{2}}\frac{\zeta_{E}^{-\vartheta_{E}}}{2\Gamma{(\vartheta_{E})}}$,
$\Xi_{8}=\sqrt{\frac{\beta_{E}}{\bar{\gamma}_{E}}\zeta_{E}^{2}}$, $\beta_{E}$
and $\vartheta_{E}$ are the shape and scale parameter of
$\mathcal{S}-\mathcal{I}_{p}-\mathcal{E}_{q}$ links, respectively and
$\bar{\gamma}_{E}$ is the average SNR.

Case-II: With the help of a similar process followed while deriving (2.3), the
PDF for NC eavesdropper can be expressed as

$\displaystyle f_{\gamma_{E,NC}}(\gamma)=$
$\displaystyle\sum_{\Lambda_{5}=0}^{\infty}\sum_{r_{1}+...+r_{(\vartheta_{E}-1)}=\Lambda_{5}}\Xi_{10}\gamma^{\frac{\vartheta_{E}}{2}-1+\Xi_{9}}$
$\displaystyle\times$ $\displaystyle
e^{-(\Xi_{8}+\Lambda_{5}\Xi_{8})\sqrt{\gamma}},$ (22)

where $\Xi_{9}=\sum_{\Lambda_{4}}\frac{\Lambda_{4}r_{\Lambda_{4}}}{2}$ and
$\Xi_{10}=\binom{L-1}{\Lambda_{5}}\binom{\Lambda_{5}}{r_{1}...r_{\vartheta_{E-1}}}(\frac{\Xi_{8}^{\Lambda_{4}}}{\Lambda_{4}!})^{\Lambda_{5}}(-1)^{\Lambda_{5}}\mathcal{L}\Xi_{7}$.
Note that assuming $\beta_{E}=\beta_{d}$, $\vartheta_{E}=\vartheta_{d}$,
$\zeta_{E}=\zeta_{d}$, $\bar{\gamma}_{E}=\bar{\gamma}_{d}$, and
$\mathcal{N}_{E}=\mathcal{N}_{d}$ the Scenerio-II reduces to Scenerio-III.

$\displaystyle
SOP_{\gamma_{e,c}}^{(\infty)}=\sum_{\kappa_{1}=1}^{2\alpha_{e}}\Xi_{2}\Xi_{3}\mathcal{A}_{1}^{\omega_{1}-1}\mathcal{A}_{2}^{\mathcal{S}_{1,\kappa_{1}}-1}\alpha_{e}^{\omega_{0}-\omega_{1}-1}\Psi^{-\alpha_{e}L\mu_{e}}\frac{\prod_{p_{1}=1;p_{1}\neq\kappa_{1}}^{2\alpha_{e}}\Gamma(\mathcal{S}_{1,\kappa_{1}}-\mathcal{S}_{1,p_{1}})\prod_{p_{1}=1}^{1+2\alpha_{e}}\Gamma(1+\mathcal{S}_{2,p_{1}}-\mathcal{S}_{1,\kappa_{1}})}{\prod_{p_{1}=1+2\alpha_{e}}^{3\alpha_{e}}\Gamma(1+\mathcal{S}_{1,p_{1}}-\mathcal{S}_{1,\kappa_{1}})}.$
(25) $\displaystyle SOP_{\gamma_{e,nc}}^{(\infty)}$
$\displaystyle=\sum_{k_{1}+...+k_{\mu_{e}-1}=\Lambda_{3}}\sum_{\Lambda_{3}=0}^{\infty}\sum_{\kappa_{2}=1}^{2}\frac{2\Xi_{5}\Xi_{2}\sqrt{B_{2}\pi}\prod_{p_{2}=1;p_{2}\neq\kappa_{2}}^{2}\Gamma(\mathcal{S}_{3,\kappa_{2}}-\mathcal{S}_{3,p_{2}})\prod_{p_{2}=1}^{2\mathcal{B}_{2}+\mathcal{B}_{1}+1}\Gamma(1+\mathcal{S}_{4,p_{2}}-\mathcal{S}_{3,\kappa_{2}})}{(2\pi)^{B_{2}}\Psi^{(\frac{\alpha_{e}\mu_{e}}{2}+\Xi_{6})}\mathcal{A}_{1}^{(\frac{\alpha_{e}\mu_{e}+\vartheta_{d}}{2}+\Xi_{6})}\mathcal{A}_{3}^{-\mathcal{S}_{3,\kappa_{2}}+1}\Gamma(1+\mathcal{S}_{3,3}-\mathcal{S}_{3,\kappa_{2}})}.$
(26) $\displaystyle
SOP_{\gamma_{E,C}}^{(\infty)}=\sum_{\kappa_{3}=1}^{2}\Xi_{2}\Xi_{7}\mathcal{A}_{1}^{\omega_{2}-1}\mathcal{A}_{4}^{\mathcal{S}_{5,\kappa_{3}}-1}\pi^{-\frac{1}{2}}\Psi^{-\frac{\vartheta_{E}}{2}}\frac{\prod_{p_{3}=1;p_{3}\neq\kappa_{3}}^{2}\Gamma(\mathcal{S}_{5,\kappa_{3}}-\mathcal{S}_{5,p_{3}})\prod_{p_{3}=1}^{\mathcal{B}_{1}+3}\Gamma(1+\mathcal{S}_{6,p_{3}}-\mathcal{S}_{5,\kappa_{3}})}{\Gamma(1+\mathcal{S}_{5,3}-\mathcal{S}_{5,\kappa_{3}})}.$
(27) $\displaystyle
SOP_{\gamma_{E,NC}}^{(\infty)}=\sum_{\Lambda_{5}=0}^{\infty}\sum_{r_{1}+...+r_{(\vartheta_{E}-1)=\Lambda_{5}}}\sum_{\kappa_{4}=1}^{2}\frac{\Xi_{10}\Xi_{2}\prod_{p_{4}=1;p_{4}\neq\kappa_{4}}^{2}\Gamma(\mathcal{S}_{7,\kappa_{4}}-\mathcal{S}_{7,p_{4}})\prod_{p_{4}=1}^{\mathcal{B}_{1}+3}\Gamma(1+\mathcal{S}_{8,p_{4}}-\mathcal{S}_{7,\kappa_{4}})}{\sqrt{\pi}\Psi^{\frac{\vartheta_{E}}{2}+\Xi_{9}}\mathcal{A}_{1}^{-\omega_{2}+\Xi_{9}+1}\mathcal{A}_{5}^{-\mathcal{S}_{7,\kappa_{4}}+1}\Gamma(1+\mathcal{S}_{7,3}-\mathcal{S}_{7,\kappa_{4}})}.$
(28)

##  3 Performance Analysis

In this section, the novel analytical expressions of SOP are demonstrated by
considering colluding and NC eavesdropping attacks.

###  3.1 SOP Analysis

To achieve secure communication, the instantaneous secrecy capacity ($C_{s}$),
must be greater than the predetermined target secrecy rate ($\tau_{0}$),
otherwise, an SOP event occurs. Mathematically, the lower bound of SOP can be
written as

$\displaystyle SOP\geq
SOP_{L}=\int_{0}^{\infty}F(\Psi\gamma)f(\gamma)d\gamma.$ (19)

where $\Psi=2^{\tau_{0}}$.
Scenario-I:Case-I: Substituting (12) and (15) into (19), we obtain

$\displaystyle SOP_{\gamma_{e,c}}=$
$\displaystyle\Xi_{2}\Xi_{3}\Psi^{\frac{\vartheta_{d}}{2}}\int_{0}^{\infty}\gamma^{-\omega_{1}}G_{0,1}^{1,0}\left[\Xi_{4}\gamma^{a_{e}}\bigg{|}\begin{array}[]{c}-\\\
0\end{array}\right]$ (22) $\displaystyle\times$ $\displaystyle
G_{\mathcal{B}_{1}+1,3}^{2,\mathcal{B}_{1}+1}\left[\mathcal{A}_{1}\gamma\Psi\bigg{|}\begin{array}[]{c}-\frac{\vartheta_{d}}{2}+1,-\omega_{0}\\\
0,\frac{1}{2},-\frac{\vartheta_{d}}{2}\end{array}\right]d\gamma.$ (25)

Utilizing the identity [14, Eq. (2.24.1.1)] and performing some mathematical
manipulations, the lower bound of SOP is expressed as

$\displaystyle SOP_{\gamma_{e,c}}=$
$\displaystyle\frac{\Xi_{2}\Xi_{3}\mathcal{A}_{1}^{\omega_{1}-1}}{\alpha_{e}^{1-\omega_{0}+\omega_{1}}\Psi^{\alpha_{e}\mathcal{L}\mu_{e}}}G_{3\alpha_{e},1+2\alpha_{e}}^{1+2\alpha_{e},2\alpha_{e}}\left[\mathcal{A}_{2}\bigg{|}\begin{array}[]{c}\mathcal{S}_{1}\\\
\mathcal{S}_{2}\end{array}\right],$ (28)

where
$\mathcal{A}_{2}=\frac{\Xi_{4}\alpha_{e}^{\alpha_{e}}}{(\mathcal{A}_{1}\Psi)^{\alpha_{e}}}$,
$\omega_{1}=-\frac{\vartheta_{d}}{2}-\mathcal{L}\alpha_{e}\mu_{e}+1$,
$t=-L\alpha_{e}\mu_{e}+1$,
$\mathcal{S}_{1}=\triangle(\alpha_{e},\omega_{1}),\triangle(\alpha_{e},\omega_{1}-\frac{1}{2}),\triangle(\alpha_{e},t)$,
and
$\mathcal{S}_{2}=0,\triangle(\alpha_{e},-\mathcal{L}\alpha_{e}\mu_{e}),\triangle(\alpha_{e},\omega_{1}+\omega_{0})$.

Case-II: By plugging (12) and (2.3) into (19) and utilizing the identity of
[14, Eq. (2.24.1.1)], the SOP for NC eavesdroppers can be obtained as

$\displaystyle
SOP_{\gamma_{e,nc}}=\sum_{k_{1}+...+k_{\mu_{e}-1}=\Lambda_{3}}\sum_{\Lambda_{3}=0}^{\infty}2\Xi_{5}\Xi_{2}\sqrt{\mathcal{B}_{2}\pi}(2\pi)^{-\mathcal{B}_{2}}$
$\displaystyle\times\frac{\Psi^{-(\frac{\alpha_{e}\mu_{e}}{2}+\Xi_{6})}}{\mathcal{A}_{1}^{(\frac{\alpha_{e}\mu_{e}+\vartheta_{d}}{2}+\Xi_{6})}}G_{3,2\mathcal{B}_{2}+\mathcal{B}_{1}+1}^{2\mathcal{B}_{2}+\mathcal{B}_{1}+1,2}\left[\mathcal{A}_{3}\bigg{|}\begin{array}[]{c}\mathcal{S}_{3}\\\
\mathcal{S}_{4}\end{array}\right],$ (31)

where
$\mathcal{A}_{3}=\frac{[(1+\Lambda_{3})\mu_{e}\bar{\gamma}^{-\frac{\alpha_{e}}{2}}]^{2\mathcal{B}_{2}}}{(2\mathcal{B}_{2})^{2\mathcal{B}_{2}}\mathcal{A}_{1}\Psi},\mathcal{B}_{2}=\frac{1}{\alpha_{e}},\mathcal{S}_{3}=\triangle(1,-\frac{\vartheta_{d}}{2}-\frac{\alpha_{e}\mu_{e}}{2}+1-\Xi_{6})$,
and
$\mathcal{S}_{4}=\triangle(2\mathcal{B}_{2},0),\triangle(1,-\frac{\alpha_{e}\mu_{e}}{2}-\Xi_{6})$.
Scenerio-II:Case-I: Substituting (12) and (20) into (19) and following the
similar process of (3.1), we obtain

$\displaystyle SOP_{\gamma_{E,C}}=$
$\displaystyle\frac{\Xi_{2}\Xi_{7}\mathcal{A}_{1}^{\omega_{2}-1}}{\pi^{\frac{1}{2}}\Psi^{\frac{\vartheta_{E}}{2}}}G_{3,\mathcal{B}_{1}+3}^{\mathcal{B}_{1}+3,2}\left[\mathcal{A}_{4}\bigg{|}\begin{array}[]{c}\mathcal{S}_{5}\\\
\mathcal{S}_{6}\end{array}\right],$ (34)

where $\mathcal{A}_{4}=\frac{\Xi_{8}^{2}}{4\mathcal{A}_{1}\Psi}$,
$\omega_{2}=-\frac{\vartheta_{d}}{2}-\frac{\vartheta_{E}}{2}+1$,
$\mathcal{S}_{5}=\triangle(1,\omega_{2})$, and
$\mathcal{S}_{6}=\triangle(2,0),\triangle(1,-\frac{\vartheta_{E}}{2})$.
Case-II: Similar to (34), replacing (12) and (2.3) into (19), the lower bound
of SOP for NC eavesdroppers with RIS can be obtained as

$\displaystyle SOP_{\gamma_{E,NC}}=$
$\displaystyle\sum_{\Lambda_{5}=0}^{\infty}\sum_{r_{1}+...+r_{(\vartheta_{E}-1)=\Lambda_{5}}}\frac{\Xi_{10}\Xi_{2}}{\sqrt{\pi}\Psi^{\frac{\vartheta_{E}}{2}+\Xi_{9}}}\mathcal{A}_{1}^{\omega_{2}-\Xi_{9}-1}$
$\displaystyle\times$ $\displaystyle
G_{3,\mathcal{B}_{1}+3}^{\mathcal{B}_{1}+3,2}\left[\mathcal{A}_{5}\bigg{|}\begin{array}[]{c}\mathcal{S}_{7}\\\
\mathcal{S}_{8}\end{array}\right],$ (37)

where
$\mathcal{A}_{5}=\frac{(\Xi_{8}+\Lambda_{5}\Xi_{8})^{2}}{4\Psi\mathcal{A}_{1}}$,
$\mathcal{S}_{7}=\triangle(1,\omega_{2}-\Xi_{9})$, and
$\mathcal{S}_{8}=\triangle(2,0),\triangle(1,-\frac{\vartheta_{E}}{2}-\Xi_{9})$.

###  3.2 Asymptotic Analysis

The exact SOP in section A does not provide much insight into the secure
outage performance of the proposed model. Hence, we also derive the asymptotic
expression of SOP in this section. Herein, Meijer’s $G$ term is converted into
an asymptotic term by utilizing [16, Eq. (41)] as expressed in (25), (26),
(27), and (28).

##  4 Numerical Results

Figure 1: SOP vs $\bar{\gamma}_{d}$ for the selected values of
$\mathcal{N}_{E}$ with $\mathcal{M}=20$.

In this section, the effects of the system parameters are presented through
graphs generated using analytical expressions. The validity of each graph is
confirmed through 100,000 random channel sample simulations performed in
MATLAB. The system parameters are set to
$\alpha_{s}=\alpha_{r}=2,\alpha_{i}=\alpha_{e}=1$,
$\mu_{s}=\mu_{r}=\mu_{i}=\mu_{e}=1$, $\mathcal{N}_{d}=\mathcal{N}_{E}=2$,
$\mathcal{L}=2$, $\mathcal{M}=2$, $\beta_{d}=\beta_{E}=0.5$,
$\tau_{0}=0.1$,$\Omega_{s}=\Omega_{r}=1$,$\bar{\gamma}_{i}=100$ dB, and
$\bar{\gamma}_{e}=\bar{\gamma}_{E}=1$ dB, unless specified otherwise.
Additionally, asymptotic analysis is conducted showing a tight match at a high
SNR regime with the analytical one. Furthermore, the correctness of the
derived expressions is confirmed by close matches between analytical results
and simulation results. Note that in all figures, ”Sim” indicates the
simulation results.

Figure 1 illustrates the effects of interference and $\mathcal{N}_{E}$ on
secure outage performance. As is evident, the security performance worsens
with interference and an increase in the number of reflecting elements in
$\mathcal{I}$. This occurs because interference reduces the SINR at
$\mathcal{D}$ and the eavesdropper link becomes stronger as $\mathcal{N}_{E}$
increases.

Figure 2: SOP vs $\bar{\gamma}_{d}$ for the selected values of $\mathcal{M}$
and $\bar{\gamma}_{e}$.

The illustration in Fig. 2 demonstrates the impact of the number of
interferers at $\mathcal{D}$. It is clear that as the number of $\mathcal{M}$
increases, the SOP declines due to the increase in interference that lowers
the SINR at $\mathcal{D}$. Additionally, a rise in $\bar{\gamma}_{e}$
strengthens the eavesdropper link, resulting in a higher secrecy outage.

Figure 3: SOP vs $\bar{\gamma}_{d}$ for the selected values of
$\mathcal{N}_{d}$ with $\mathcal{N}_{E}=20$.

Figure 3 demonstrates the significance of RIS and reflecting elements in
$\mathcal{R}$ showing that a degraded secrecy performance is achieved when RIS
is present in the eavesdropper link as opposed to a direct link without RIS.
Additionally, as the number of $\mathcal{N}_{d}$ increases, there is an
improvement in the SNR at $\mathcal{D}$, accompanied by a reduction in the
SOP.

Figure 4 depicts the impact of the number of eavesdroppers along with the
multiple eavesdroppers’ collusion and non-collusion.

Figure 4: SOP vs $\bar{\gamma}_{d}$ for the selected values of $\mathcal{L}$.

As can be observed, the SOP rises with an increase in $\mathcal{L}$. The
impact of colluding eavesdroppers is observed as more harmful since they are
able to collaborate and combine their resources and abilities to intercept
confidential information.

##  5 Conclusions

This study focuses on the secrecy outage performance of a RIS-aided wireless
system in the presence of multiple eavesdroppers and interferers. All the
results are justified by Monte-Carlo simulations and asymptotic analysis is
also carried out at high SNR regime. The numerical results indicate that the
impact of colluding eavesdroppers is more severe than that of non-colluding
eavesdroppers. Moreover, the presence of interferers also imposes a
detrimental effect on SOP, which can be reduced by adjusting the number of
reflecting elements of both RIS.

## References

  * [1] T. Hossain, S. Shabab, A. S. M. Badrudduza, M. K. Kundu, and I. S. Ansari, “On the physical layer security performance over RIS-aided dual-hop RF-UOWC mixed network,” _IEEE Transactions on Vehicular Technology_ , pp. 1–12, 2022\.
  * [2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” _IEEE Communications Magazine_ , vol. 58, no. 1, pp. 106–112, 2019.
  * [3] A. Badrudduza, M. Ibrahim, S. R. Islam, M. S. Hossen, M. K. Kundu, I. S. Ansari, and H. Yu, “Security at the physical layer over GG fading and mEGG turbulence induced RF-UOWC mixed system,” _IEEE Access_ , vol. 9, pp. 18 123–18 136, 2021.
  * [4] L. Yang, J. Yang, W. Xie, M. O. Hasna, T. Tsiftsis, and M. Di Renzo, “Secrecy performance analysis of RIS-aided wireless communication systems,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 10, pp. 12 296–12 300, 2020.
  * [5] L. Wei, K. Wang, C. Pan, and M. Elkashlan, “Secrecy performance analysis of RIS-aided communication system with randomly flying eavesdroppers,” _IEEE Wireless Communications Letters_ , vol. 11, no. 10, pp. 2240–2244, 2022\.
  * [6] A. K. Yadav, S. Yadav, A. Pandey, and A. Silva, “On the secrecy performance of RIS-enabled wireless communications over Nakagami-$m$ fading channels,” _ICT Express_ , 2022.
  * [7] Z. Zhang, J. Chen, Y. Liu, Q. Wu, B. He, and L. Yang, “On the secrecy design of STAR-RIS assisted uplink NOMA networks,” _IEEE Transactions on Wireless Communications_ , vol. 21, no. 12, pp. 11 207–11 221, 2022.
  * [8] Z. Tang, T. Hou, Y. Liu, J. Zhang, and C. Zhong, “A novel design of RIS for enhancing the physical layer security for RIS-aided NOMA networks,” _IEEE Wireless Communications Letters_ , vol. 10, no. 11, pp. 2398–2401, 2021\.
  * [9] M. Tania, M. Ibrahim, M. Hossen, A. Badrudduza, and M. Kundu, “Combined impacts of co-channel interference and correlation on secrecy performance over $\kappa$-$\mu$ shadowed fading channel,” in _2022 International Conference on Advancement in Electrical and Electronic Engineering (ICAEEE)_. IEEE, 2022, pp. 1–6.
  * [10] B. Li, Y. Zou, J. Zhu, C. Shi, and H. Zeng, “Secrecy outage analysis of relay aided multiuser scheduling for wireless networks in the face of co-channel interference,” _IEEE Transactions on Vehicular Technology_ , 2022.
  * [11] T. Ssettumba, A. H. Abd El-Malek, M. Elsabrouty, and M. Abo-Zahhad, “Physical layer security enhancement for internet of things in the presence of co-channel interference and multiple eavesdroppers,” _IEEE Internet of Things Journal_ , vol. 6, no. 4, pp. 6441–6452, 2019.
  * [12] I. S. Gradshteyn and I. M. Ryzhik, _Table of integrals, series, and products_. Academic press, 2014.
  * [13] S. Primak, V. Kontorovich, and V. Lyandres, _Stochastic methods and their applications to communications: stochastic differential equations approach_. John Wiley & Sons, 2005.
  * [14] P. Prudnikov, Anatoli, Y. A. Brychkov, O. I. Marichev, and R. H. Romer, “Integrals and series,” 1988.
  * [15] N. H. Juel, A. Badrudduza, S. R. Islam, S. H. Islam, M. K. Kundu, I. S. Ansari, M. M. Mowla, and K.-S. Kwak, “Secrecy performance analysis of mixed $\alpha$\- $\mu$ and exponentiated Weibull RF-FSO cooperative relaying system,” _IEEE Access_ , vol. 9, pp. 72 342–72 356, 2021.
  * [16] I. S. Ansari, F. Yilmaz, and M.-S. Alouini, “Performance analysis of free-space optical links over Málaga ($\mathcal{M}$) turbulence channels with pointing errors,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 1, pp. 91–102, 2015.


# Designable exciton mixing through layer alignment in WS2-graphene
heterostructures

Amir Kleiner  Department of Molecular Chemistry and Materials Science,
Weizmann Institute of Science, Rehovot 7610001, Israel  Daniel Hernangómez-
Pérez  Department of Molecular Chemistry and Materials Science, Weizmann
Institute of Science, Rehovot 7610001, Israel  Sivan Refaely-Abramson
Department of Molecular Chemistry and Materials Science, Weizmann Institute of
Science, Rehovot 7610001, Israel<EMAIL_ADDRESS>
###### Abstract

Optical properties of heterostructures composed of layered 2D materials, such
as transition metal dichalcogenides (TMDs) and graphene, are broadly explored.
Of particular interest are light-induced energy transfer mechanisms in these
materials and their structural roots. Here, we use state-of-the-art first-
principles calculations to study the excitonic composition and the absorption
properties of WS2–graphene heterostructures as a function of interlayer
alignment and the local strain resulting from it. We find that Brillouin zone
mismatch and the associated energy level alignment between the graphene Dirac
cone and the TMD bands dictate an interplay between interlayer and intralayer
excitons, mixing together in the many-body representation upon the strain-
induced symmetry breaking in the interacting layers. Examining the
representative cases of the 0∘ and 30∘ interlayer twist angles, we find that
this exciton mixing strongly varies as a function of the relative alignment.
We quantify the effect of these structural modifications on exciton charge
separation between the layers and the associated graphene-induced homogeneous
broadening of the absorption resonances. Our findings provide guidelines for
controllable optical excitations upon interface design and shed light on the
importance of many-body effects in the understanding of optical phenomena in
complex heterostructures.

transition-metal dichalcogenides, graphene, excitons, heterostructures, many-
body perturbation theory, GW-BSE

##  I Introduction

Two-dimensional layered van der Waals heterostructures have been extensively
studied for a wealth of applications in photophysics. Their structural
complexity provides adjustable and designable light-matter features, offering
the generation of controllable optical states for application in
optoelectronics, light harvesting, and material-based information science. [1,
2, 3, 4, 5, 6, 7]. Of particular interest are layered heterostructures (HSs)
composed of graphene (Gr) and transition metal dichalcogenides (TMDs) [8, 9,
10, 11]. These interfaces combine the semi-metallic nature, large charge
carrier mobility, and constant infrared optical absorption of Gr [12, 13, 14,
15, 16, 17] with the semiconducting nature of TMDs, characterized by strong
exciton binding and significant spin-orbit coupling [18, 19, 20, 21, 22, 23].
Reflectance contrast measurements in these interfaces show line broadening [9]
that can be assigned to charge transfer processes between the coupled layers
[24, 25, 26, 27]. Recent time-resolved angle-resolved photoemission
spectroscopy (tr-ARPES) studies further suggest a relation between interlayer
coupling and reciprocal-space spreading of the optical transitions [28]. These
processes are particularly affected by intrinsic atomic reconstruction upon
interlayer twisting [29, 30].

From a theoretical point of view, Gr-TMD HSs serve as a fascinating test case
to establish a direct relation between the intrinsic structural disorder
induced by layer stacking, which is sensitive to interlayer alignment, and the
observed optical signature and spectral broadening. The presence of the Gr
Dirac cone within the TMD band gap [31, 32, 33] opens pathways to the
tunability of the electronic structure upon interface alignment and layer
thickness, well known to modify the magnitude and directness of the TMD band
gap [34, 35, 36, 37]. The effect of Gr on the TMD electronic levels is
typically associated with non-local dielectric screening, manifested in band
gap renormalization [38, 39, 40] and subsequent reduction in the excitonic
binding energies [41]. However, a first-principles understanding of the change
in the exciton states themselves upon layer interfacing is yet to be achieved.

Recent studies used many-body perturbation theory to show that upon TMD-TMD
heterostructure composition, excitons include a wealth of electron-hole
transitions with varying spin and momentum properties [42, 43, 44], leading to
the emergence of hybridized excitons with mixed interlayer and intralayer
nature. Such state mixing results from local symmetry breaking and resonating
excitation energies, manifesting many-body effects associated with the
relative reciprocal space alignment of the layers, as well as the interface-
induced atomic reconstruction and the associated local strain [45, 46].
Recently we have explored the effect of atomic defects on the optical
properties in Gr-TMD HSs [47], finding that largely mixed subgap, defect-
induced excitons dominate the absorption, a property that in the monolayer TMD
case leads to greatly reduced valley selectivity [48, 49]. The optical
activity of Gr-TMD junctions is thus closely coupled to their interfacial
structural details through the many-body exciton nature. A first-principles
understanding of the exciton state mixing as a function of interface layer
alignment in Gr-TMD HSs is then of great interest and can shed light on the
underlying mechanisms dominating the excited-state phenomena in these systems.

In this work, we study the effect of layer alignment on optical and excitonic
properties in WS2–graphene heterobilayers using many-body perturbation theory.
Exploring two representative configurations of this interface, with both 0∘
and 30∘ interlayer twist angles, we show that optical excitations largely mix
electron-hole transitions within the TMD and Gr layers and between them. This
mixing depends on the interfacial configuration through the angle-dependent
band alignment. As a result, states that are optically active due to their TMD
layer contribution also have significant contributions from the Gr layer. We
analyze the exciton charge separation associated with this effect and
investigate the absorption line broadening resulting from the large amount of
excitons contributing to the optical resonances. Our findings suggest that
charge transfer mechanisms in these heterostructures are largely tunable with
the twist angle, with significant charge separation occurring already upon
light absorption and setting a coherent starting point to charge transfer
dynamics, e.g., by efficient scattering into graphene-induced low-lying
states.

##  II Results and Discussion

Figure 1: (a,b) Atomic structure of the studied WS2-Gr HS with $0^{\circ}$ and
$30^{\circ}$ interlayer twist angles (left); periodic supercell is shown in
grey, with the corresponding BZs (right) of WS2 (red) and graphene (blue) and
the mini-BZ of the supercell (black). (c,d) DFT band structures of the
$0^{\circ}$ and $30^{\circ}$ twisted HS (left) and the computed GW energy
levels (right) of the TMD gap and its alignment with the graphene Dirac point.
The TMD band gap and the TMD-graphene energy gap are shown in eV. The color
code represents the layer contribution to each band, ranging from pure TMD
(red) to pure graphene (purple).

The studied heterostructures are presented in Fig. 1(a,b). We use commensurate
periodic supercells with $4\times 4$ repeating elementary units of the TMD
layer and $5\times 5$ repeating elementary units of the Gr layer for the
$0^{\circ}$-twist angle; and $3\sqrt{3}\times 3\sqrt{3}$ repeating elementary
units of the Gr Layer for the $30^{\circ}$-twist angle (Fig. 1(a,b), left).
These alignments result in a minimal strain of $2.6\%$ for the
$0^{\circ}$-twist angle and $1.3\%$ for the $30^{\circ}$-twist angle in the Gr
layer, respectively, and no strain in the TMD layer. However, local strain is
induced in the TMD upon structural relaxation as a function of the Gr-TMD
layer alignment (see SI for full details). The unit cell Brillouin zones (BZ),
corresponding to the separate layers, fold in this configuration onto the
smaller, shared mini-BZ (mBZ) of the supercell (Fig. 1(a,b), right). The TMD
bands fold identically in both configurations (due to the equivalent TMD-layer
BZ in both cases), resulting in the TMD minimal gap at the $K$ point (or
equivalently $K^{\prime}$; we consider only $K$ in this work for brevity). The
conduction-band minimum is at the $\Lambda$ point of the unit cell (located
midway between $K$ and $\Gamma$), folded onto the $K$ point of the mBZ. The
different twist configurations lead to different folding of the graphene bands
in the mBZ of the two HSs: while in the $0^{\circ}$ HS the Dirac point folds
to the mBZ $K$ point, in the $30^{\circ}$ HS, it folds to the mBZ $\Gamma$
point. As we show below, this enforces substantial differences in the allowed
optical transitions for both systems.

We use density functional theory (DFT)[50] to calculate the electronic band
structure of the two systems, as shown in the left panels of Fig. 1(c,d) along
the $\Gamma$-$K$-$M$-$\Gamma$ path of the mBZ. Band colors give the relative
contribution of the graphene layer (purple) and the TMD layer (red), with some
states hybridized at band crossings (see SI for further details). In
completion with the BZ folding scheme, in the case of the $0^{\circ}$-twist HS
(Fig. 1(c)), the graphene Dirac cone is aligned in reciprocal space with the
TMD direct band gap, both folded onto the K point of the mBZ. For the
$30^{\circ}$-twist HS (Fig. 1(d)), the Dirac cone is centered at the $\Gamma$
point, and the minimal gap of the TMD occurs at the $K$ point of the mBZ.
Importantly, the DFT electronic energy-level alignment of the Dirac point
within the gap is different in the two systems, located $0.89$ eV and $0.51$
eV below the TMD conduction band minimum at $K$ for $0^{\circ}$ and
$30^{\circ}$-twisted HSs, respectively.

Figure 2:  (a,b) Absorbance spectra of the $0^{\circ}$ and $30^{\circ}$
twisted HSs (black line). The binding energies of each optically active
exciton are shown with dots, with colors representing the layer contribution
of each excitation, from pure TMD (red) to pure graphene (purple). (c,d)
Momentum-space representation of selected excitons, corresponding to the
shaded vertical grey lines in panels (a,b). Lower and upper panels show the
hole and electron distribution in the mBZ, respectively.

We further apply many-body perturbation theory within the G0W0 approximation
[51], using the DFT electronic states and energies as a starting point, to
obtain corrected quasiparticle energy levels as shown in the right panels of
Fig. 1(c,d). The resulting TMD GW band gaps are $2.59$ eV in both systems. Our
computed GW gaps are $\sim 10-20\%$ larger than the experimental band gaps
reported for WS2–Gr heterobilayers, of $2.1-2.3$ eV [52, 53, 26, 24, 54, 55,
25, 10, 40]. While still within a reasonable error bar, this difference is
larger than expected. We assign it to both further surface renormalization in
experiment, as well as to a computational effect of state hybridization in the
mBZ, as was recently observed in other systems as well [45]. The difference in
the Dirac cone alignment within the TMD gap remains, now located $1.50$ eV and
$1.04$ eV below the TMD conduction band minimum at K, for $0^{\circ}$ and
$30^{\circ}$-twisted HSs, respectively. We note recent experimental
observations supporting the Gr Fermi level being closer to the conduction
region [24, 11]. To evaluate the graphene-induced change in the band gap, we
compare our results with a similar GW calculation of an isolated TMD layer
that includes the atomic reconstruction due to graphene. This results in a gap
renormalization of $\sim$0.3 eV in both systems (see SI), in good agreement
with experiment [24, 26, 52, 56, 25].

Next, we explore the optical spectra and excitonic states of the examined HSs
by employing the Bethe-Salpeter equation (BSE) [57, 51] within many-body
perturbation theory. Fig. 2(a) shows the calculated absorbance of the
$0^{\circ}$ HS (black curve). On top of the absorbance, we show the binding
energies of all the excitons in this energy range with finite oscillator
strength ($\mu_{X}$, here chosen as $\mu_{X}$>$1$). The binding energy is
computed by subtracting the energy of each exciton ($\Omega_{X}$) from the
quasiparticle energy gaps $E_{c\mathbf{k}}-E_{v\mathbf{k}}$, considering all
the electron ($v$) - hole ($c$) transitions composing the exciton state at
each k-point, weighted by the exciton coefficients obtained from the solution
of the BSE ($A^{X}_{vc\mathbf{k}}$) and summed over all the BZ, through
$E^{X}_{\textnormal{bind}}=\sum_{vc\mathbf{k}}|A^{X}_{vc\mathbf{k}}|^{2}(E_{c\mathbf{k}}-E_{v\mathbf{k}})-\Omega_{X}$
(see SI for full details). Dot colors represent the relative contribution of
each layer to each exciton: intralayer excitons located solely at the graphene
layer (purple), intralayer excitons located solely at the TMD layer (red), and
excitons delocalized between both layers (ranging in between). For the
$0^{\circ}$ twisted HS, the main excitons composing the absorbance peaks have
binding energies $\sim$100 meV, reduced in comparison to the case of pristine
TMDs, as was also observed before [41, 26, 58, 40]. Transitions at lower
energies oscillate around the constant absorbance of graphene at $2.4\%$ [59,
60, 15, 14, 38, 47], and are primarily composed of intralayer graphene states.
The two pronounced peaks at $2.49$ eV and at $2.82$ eV, referred as $\bar{A}$
and $\bar{B}$, are mainly intralayer TMD excitons, with a slight mixing of
graphene states. These states are associated with the well-known $A$ and $B$
peaks of TMDs, here also mixing additional graphene-containing transitions, as
well as transitions corresponding to the excitation of the TMD $\Lambda$
point, in agreement with recent experimental observations [28].

Interestingly, in the case of the $30^{\circ}$-twist HS (Fig. 2(b), our
results show a very different absorbance composition. The low-energy states
oscillate below the graphene constant absorption value in the infrared optical
range and consequently have significant interlayer contributions. We observe
the main TMD-like $\bar{A},\bar{B}$ peak features at similar energies of
$2.50$ eV and $2.83$ eV. The substantially reduced absorbance between the low-
energy region and the $\tilde{\textnormal{A}}$ peak points to the fully
interlayer-nature of the excitons in this region, in which intralayer graphene
contributions are no longer present while intralayer TMD contributions are not
yet present, thus decreasing the oscillator strength of these low-lying
states. In both systems, we also observe dark excitons with relatively large
binding energies, but with no signature in absorbance. Additionally, some of
the strongly bound excitons exhibit a substantial contribution from the
graphene layer.

To further analyze the excitations composing the complex absorbance picture
described above, we plot the k-resolved exciton compositions in the energy
regions of selected absorbance peaks, Fig. 2(c,d). In each energy region, the
bright exciton states are shown in an interval of $\pm 5$ meV around the
maximum of the absorption peaks. We show the integrated contribution of the
holes (lower panels) and electrons (upper panels) separately, weighted by the
exciton oscillator strength (see SI for further analysis). This representation
allows further insights into the different features of the two structures.
Notably, the k-space spreading of the low-energy exciton ($X_{l}$) greatly
differs between the two examined structures. For the $0^{\circ}$ twist angle,
the involved electron-hole transitions are primarily of intralayer graphene
nature. The k-space spreading around the $K$, $K^{\prime}$ valleys is a result
of the low-lying excitation near the Dirac point in this structure. In
contrast, for the $30^{\circ}$ twist angle structure, the low-lying excitation
has an interlayer graphene-TMD nature. It is also centered around the Dirac
cone, this time at the $\Gamma$ region. This is a direct manifestation of the
electronic band-alignment effect on the optical transitions. We note in
passing that though the observed optical transitions are momentum-direct in
the mBZ, some are not expected in the unfolded BZ (UBZ) framework. This
property depends on the BZ folding associated with the specific twist angle
and was recently shown by some of us to induce unexpected optical transitions
also in twisted TMD-TMD heterobilayers [45]. The $\bar{\textnormal{A}}$ peak
region in both systems is mainly composed of intralayer TMD transitions around
the $K$, $K^{\prime}$ points of the mBZ; in both states, further mixing occurs
with intralayer graphene and interlayer TMD-graphene transitions in the
$0^{\circ}$ and $30^{\circ}$ twist structures, respectively. The
$\bar{\textnormal{B}}$ peak region shows large mixing of graphene-graphene,
TMD-TMD, graphene-TMD, and TMD-graphene transitions.

The above findings point to the strongly mixed interlayer and intralayer
nature of the excitations, which strongly depends on the band alignment
resulting from the twist angle. We note that while the mBZ is determined by
the choice of the periodic supercell for each structure, we consider the
examined structures as good test cases to emphasize these delicate structural
dependencies, even though experimental results may divert from our specific
predictions upon structural variations. The binding energies of the various
excitons manifest their intralayer TMD contribution; interlayer TMD-Gr and
intralayer Gr-Gr transitions have smaller binding energies. Importantly,
strongly mixed states induce bound excitons that hold some interlayer nature -
of great interest for charge transfer decay processes proceeding the light
excitation. In the following, we further analyze the twist-induced excitonic
nature and, in particular, its effect on exciton charge separation and
spectral broadening.

Figure 3: Measure of the interlayer charge separation for each exciton state
in the examined (a) $0^{\circ}$ and (b) $30^{\circ}$ twisted HSs. Negative
values are for excitons with holes located at the TMD layer and electrons at
the graphene layer, and positive values for opposite transitions. Dot colors
represent the overall layer contribution of each exciton state, with purple
corresponding to excitons located at the graphene layer only, red to excitons
located at the TMD layer only, and excitons delocalized in both layers spread
in between. Only excitons with oscillator strength $\mu_{X}$>$5$ are shown
(see SI for lower thresholds).

For a numerical determination of the exciton charge-separation nature, Fig. 3
shows an analysis of the interlayer electron-hole contribution per exciton in
both structures. We differentiate between excitons with electrons and holes
localized in the same layer - namely intralayer excitons, and those with
electrons and holes localized in different layers - namely interlayer excitons
(see SI). We specify the directionality of these transitions, with a
distinction between excitons with holes located on the Gr and electrons on the
TMD layer (Gr $\rightarrow$ TMD) and those with holes located on the TMD and
electrons on the Gr layer (TMD $\rightarrow$ Gr). Each dot corresponds to an
exciton state, with colors representing its overall layer contribution. The
absorption spectra are presented in the background. Our analysis shows a clear
distinction in the interlayer exciton nature between the two structures. For
the $0^{\circ}$ twist HS, Fig. 3(a), most of the low-lying excitations are of
intralayer graphene nature. The exciton charge separation mostly occurs around
the $\bar{A}$ and $\bar{B}$ peaks and spreads rather evenly in both
directions. A very different picture emerges, however, for the $30^{\circ}$
twisted HS, Fig. 3 (b), in which the charge separation - occurring already in
the low-lying excitation energies - has preferred directionality, with holes
located in the graphene layer and electrons in the TMD layer. This property
remains in the $\bar{A}$ and $\bar{B}$ peak regions. In this case, intralayer
graphene excitons barely participate in the absorption. These results
emphasize how a small change in the interlayer alignment can lead to
substantial effects on the excitonic nature - and in particular, on the charge
separation occurring already upon light absorption.

Figure 4: Spectral broadening of the $\tilde{\textnormal{A}}$ peak for the (a)
$0^{\circ}$ and (b) $30^{\circ}$ twist HSs. Both panels compare the HS peak
(blue) and the corresponding monolayer (ML) peak (black), with the
corresponding fit given by dashed lines. Absorption peaks have been normalized
to the peak maximum. Insets show the computed peak width (Pw) as a function of
the effective Lorentzian width (Lw) of each exciton resonance for the HSs
(blue triangles) and the corresponding MLs (black circles). The difference
between the two, $\Delta_{HS}$ (green squares), is a manifestation of the
total spectral broadening induced by structural disorder induced by graphene.

A further manifestation of the exciton state hybridization and mixed nature
can be observed in the absorption spectral broadening. It has been shown
experimentally that the presence of the graphene layer induces an absorption
line broadening compared to a separated TMD monolayer [9, 52, 61, 55, 40]. Our
calculations allow us to computationally analyze the broadening arising due to
the above-discussed state mixing and determine their structural origins (we
note that dynamical effects, expected to be dominant as well, are not
accounted here). Our GW-BSE calculation results with a set of discrete exciton
states and corresponding oscillator strengths. In order to construct the
absorption spectra shown, e.g., in Figs. 2(a,c), we broaden each BSE exciton
by a Lorentzian function centered at the exciton eigenvalue [62], weighted by
the corresponding oscillator strength, and with a scale parameter of $50$ meV
to mimic the experimentally measured peak widths at room temperature [18, 63,
64, 65]. Our above examinations reveal complex exciton compositions, with each
spectral feature composed of a large number of excitations. In such a case,
the actual peak width shown in the spectrum is generally larger than the
effective Lorentzian broadening numerically set for each individual excitonic
resonance. To isolate the effect of the graphene layer on the broadening, we
compare the width of the $\bar{A}$ peak in both alignments with the
corresponding TMD monolayer (namely, such that includes the atomic
reconstruction, but not the Gr layer), by fitting the peaks with the
appropriate Voigt profiles [66, 64] (see SI for full details). This comparison
points to strong dependence of the mixing-induced broadening on interlayer
alignment. For the $0^{\circ}$-twist structure, Fig. 4(a), the presence of
graphene induces additional optically active states that significantly broaden
the optical resonance. The overall peak width (Pw) depends linearly on the
input Lorentzian width (Lw) parameter, which dictates how many transitions
actually fall within the spectral peak range, as shown in Fig. 4(c).
Importantly, the graphene-induced broadening, namely the difference between
the HS and the monolayer broadening, $\Delta_{HS-ML}$, remains significant in
this case and is expected to be proportional to the intrinsic TMD linewidth
and at the order of 20-50 meV. In contrast, for the $30^{\circ}$-twist
structure, Fig. 4(b), the presence of graphene has a negligible effect on the
peak broadening (though it adds a certain degree of skewness towards higher
energies). In this case, $\Delta_{HS-ML}$ is close to zero, up to $10$meV
regardless of the Lorenzian scaling. This analysis suggests that the effect of
graphene on the spectral width has its origins associated with the exciton
nature, and points to a direct relation between the observed broadening and
the underlying structure composition and twist. More interestingly, it also
suggests that the broadening difference, $\Delta_{HS-ML}$, is a parameter that
captures the complexity of the many-body exciton hybridization.

##  III Conclusion

In conclusion, we presented a many-body first-principles study of the
excitonic properties in WS2-graphene heterostructures with two different twist
angles between the layers - $0^{\circ}$ and $30^{\circ}$. We found that the
optical response of these heterostructures is highly complex, with a
significantly hybridized nature resulting from structure-specific mixing
between interlayer and intralayer transitions. This mixing is strongly twist-
angle dependent, an outcome of the relative band alignment between the
graphene Dirac cone and the TMD band edges, as well as of the momentum
mismatch between the layers. We analyzed the nature of these transitions
through the exciton charge separation and the change in absorption line
broadening. Our results demonstrate an intriguing structural tunability of
optical excitations in layered heterostructures, serving as a designable
starting point for efficient relaxation in these systems.

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

Supporting Information

Supporting Information is available from the author.

Acknowledgements

Computations were carried out in the Chemfarm local cluster at the Weizmann
Institute of Science and the Max Planck Computing and Data Facility cluster.
A.K. and D.H.-P. acknowledge a Minerva Foundation grant 7135421. This research
has received funding from the European Research Council (ERC), Grant agreement
No.101041159, and from the German Research Foundation (DFG) through the
Collaborative Research Center SFB 1277 (Project-ID314695032, project B10).
S.R.-A.is an incumbent of the Leah Omenn Career Development Chair.

Conflict of interest

The authors declare no conflict of interest.

## References

  * Geim and Grigorieva [2013] A. K. Geim and I. V. Grigorieva, Van der Waals heterostructures, Nature 499, 419 (2013).
  * Novoselov  _et al._ [2016] K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. C. Neto, 2D materials and van der Waals heterostructures, Science 353, aac9439 (2016).
  * Liu  _et al._ [2016] Y. Liu, N. O. Weiss, X. Duan, H.-C. Cheng, Y. Huang, and X. Duan, Van der Waals heterostructures and devices, Nature Reviews Materials 1, 1 (2016).
  * Massicotte  _et al._ [2016] M. Massicotte, P. Schmidt, F. Vialla, K. G. Schädler, A. Reserbat-Plantey, K. Watanabe, T. Taniguchi, K. J. Tielrooij, and F. H. L. Koppens, Picosecond photoresponse in van der Waals heterostructures, Nature Nanotechnology 11, 42 (2016).
  * Jin  _et al._ [2018] C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and T. F. Heinz, Ultrafast dynamics in van der Waals heterostructures, Nature Nanotechnology 13, 994 (2018).
  * Kennes  _et al._ [2021] D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. N. Basov, A. N. Pasupathy, and A. Rubio, Moiré heterostructures as a condensed-matter quantum simulator, Nature Physics 17, 155 (2021).
  * Lemme  _et al._ [2022] M. C. Lemme, D. Akinwande, C. Huyghebaert, and C. Stampfer, 2D materials for future heterogeneous electronics, Nature Communications 13, 1392 (2022).
  * Jin  _et al._ [2015] C. Jin, F. A. Rasmussen, and K. S. Thygesen, Tuning the Schottky Barrier at the Graphene/MoS2 Interface by Electron Doping: Density Functional Theory and Many-Body Calculations, The Journal of Physical Chemistry C 119, 19928 (2015).
  * Hill  _et al._ [2017] H. M. Hill, A. F. Rigosi, A. Raja, A. Chernikov, C. Roquelet, and T. F. Heinz, Exciton broadening in WS2/ graphene heterostructures, Physical Review B 96, 205401 (2017).
  * Aeschlimann  _et al._ [2020] S. Aeschlimann, A. Rossi, M. Chávez-Cervantes, R. Krause, B. Arnoldi, B. Stadtmüller, M. Aeschlimann, S. Forti, F. Fabbri, C. Coletti, and I. Gierz, Direct evidence for efficient ultrafast charge separation in epitaxial WS2/graphene heterostructures, Science Advances 6, eaay0761 (2020).
  * Krause  _et al._ [2021a] R. Krause, S. Aeschlimann, M. Chávez-Cervantes, R. Perea-Causin, S. Brem, E. Malic, S. Forti, F. Fabbri, C. Coletti, and I. Gierz, Microscopic Understanding of Ultrafast Charge Transfer in van der Waals Heterostructures, Physical Review Letters 127, 276401 (2021a).
  * Novoselov  _et al._ [2004] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science 306, 666 (2004).
  * Castro Neto  _et al._ [2009] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Reviews of Modern Physics 81, 109 (2009).
  * Yang  _et al._ [2009] L. Yang, J. Deslippe, C.-H. Park, M. L. Cohen, and S. G. Louie, Excitonic Effects on the Optical Response of Graphene and Bilayer Graphene, Physical Review Letters 103, 186802 (2009).
  * Mak  _et al._ [2008] K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Measurement of the Optical Conductivity of Graphene, Physical Review Letters 101, 196405 (2008).
  * Das Sarma  _et al._ [2011] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene, Reviews of Modern Physics 83, 407 (2011).
  * Mak  _et al._ [2014] K. F. Mak, F. H. Da Jornada, K. He, J. Deslippe, N. Petrone, J. Hone, J. Shan, S. G. Louie, and T. F. Heinz, Tuning many-body interactions in graphene: The effects of doping on excitons and carrier lifetimes, Physical review letters 112, 207401 (2014).
  * Mak  _et al._ [2010] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Atomically Thin MoS2: A New Direct-Gap Semiconductor, Physical Review Letters 105, 136805 (2010).
  * Splendiani  _et al._ [2010] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y. Chim, G. Galli, and F. Wang, Emerging Photoluminescence in Monolayer MoS2, Nano Letters 10, 1271 (2010).
  * Ramasubramaniam [2012] A. Ramasubramaniam, Large excitonic effects in monolayers of molybdenum and tungsten dichalcogenides, Physical Review B 86, 115409 (2012).
  * Ugeda  _et al._ [2014] M. M. Ugeda, A. J. Bradley, S.-F. Shi, F. H. da Jornada, Y. Zhang, D. Y. Qiu, W. Ruan, S.-K. Mo, Z. Hussain, Z.-X. Shen, F. Wang, S. G. Louie, and M. F. Crommie, Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor, Nature Materials 13, 1091 (2014).
  * Chernikov  _et al._ [2014] A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, Exciton Binding Energy and Nonhydrogenic Rydberg Series in Monolayer WS2, Physical Review Letters 113, 076802 (2014).
  * Hanbicki  _et al._ [2015] A. T. Hanbicki, M. Currie, G. Kioseoglou, A. L. Friedman, and B. T. Jonker, Measurement of high exciton binding energy in the monolayer transition-metal dichalcogenides WS2 and WSe2, Solid State Communications 203, 16 (2015).
  * Song  _et al._ [2018] Z. Song, H. Zhu, W. Shi, D. Sun, and S. Ruan, Ultrafast charge transfer in graphene-WS2 Van der Waals heterostructures, Optik 174, 62 (2018).
  * Krause  _et al._ [2021b] R. Krause, M. Chávez-Cervantes, S. Aeschlimann, S. Forti, F. Fabbri, A. Rossi, C. Coletti, C. Cacho, Y. Zhang, P. E. Majchrzak, R. T. Chapman, E. Springate, and I. Gierz, Ultrafast Charge Separation in Bilayer WS2/Graphene Heterostructure Revealed by Time- and Angle-Resolved Photoemission Spectroscopy, Frontiers in Physics 9 (2021b).
  * Trovatello  _et al._ [2022] C. Trovatello, G. Piccinini, S. Forti, F. Fabbri, A. Rossi, S. De Silvestri, C. Coletti, G. Cerullo, and S. Dal Conte, Ultrafast hot carrier transfer in WS2/graphene large area heterostructures, npj 2D Materials and Applications 6, 1 (2022).
  * Fu  _et al._ [2021] S. Fu, I. du Fossé, X. Jia, J. Xu, X. Yu, H. Zhang, W. Zheng, S. Krasel, Z. Chen, Z. M. Wang, K.-J. Tielrooij, M. Bonn, A. J. Houtepen, and H. I. Wang, Long-lived charge separation following pump-wavelength–dependent ultrafast charge transfer in graphene/WS2 heterostructures, Science Advances 7, eabd9061 (2021).
  * Dong  _et al._ [2023] S. Dong, S. Beaulieu, M. Selig, P. Rosenzweig, D. Christiansen, T. Pincelli, M. Dendzik, J. D. Ziegler, J. Maklar, R. P. Xian, A. Neef, A. Mohammed, A. Schulz, M. Stadler, M. Jetter, P. Michler, T. Taniguchi, K. Watanabe, H. Takagi, U. Starke, A. Chernikov, M. Wolf, H. Nakamura, A. Knorr, L. Rettig, and R. Ernstorfer, Observation of ultrafast interfacial Meitner-Auger energy transfer in a Van der Waals heterostructure, Nature Communications 14, 5057 (2023).
  * Weston  _et al._ [2020] A. Weston, Y. Zou, V. Enaldiev, A. Summerfield, N. Clark, V. Zólyomi, A. Graham, C. Yelgel, S. Magorrian, M. Zhou, J. Zultak, D. Hopkinson, A. Barinov, T. H. Bointon, A. Kretinin, N. R. Wilson, P. H. Beton, V. I. Fal’ko, S. J. Haigh, and R. Gorbachev, Atomic reconstruction in twisted bilayers of transition metal dichalcogenides, Nature Nanotechnology 15, 592 (2020).
  * Rosenberger  _et al._ [2020] M. R. Rosenberger, H.-J. Chuang, M. Phillips, V. P. Oleshko, K. M. McCreary, S. V. Sivaram, C. S. Hellberg, and B. T. Jonker, Twist Angle-Dependent Atomic Reconstruction and Moiré Patterns in Transition Metal Dichalcogenide Heterostructures, ACS Nano 14, 4550 (2020).
  * Pierucci  _et al._ [2016] D. Pierucci, H. Henck, J. Avila, A. Balan, C. H. Naylor, G. Patriarche, Y. J. Dappe, M. G. Silly, F. Sirotti, A. T. C. Johnson, M. C. Asensio, and A. Ouerghi, Band Alignment and Minigaps in Monolayer MoS2-Graphene van der Waals Heterostructures, Nano Letters 16, 4054 (2016).
  * Wilson  _et al._ [2017] N. R. Wilson, P. V. Nguyen, K. Seyler, P. Rivera, A. J. Marsden, Z. P. L. Laker, G. C. Constantinescu, V. Kandyba, A. Barinov, N. D. M. Hine, X. Xu, and D. H. Cobden, Determination of band offsets, hybridization, and exciton binding in 2D semiconductor heterostructures, Science Advances 3, e1601832 (2017).
  * Henck  _et al._ [2018] H. Henck, Z. Ben Aziza, D. Pierucci, F. Laourine, F. Reale, P. Palczynski, J. Chaste, M. G. Silly, F. Bertran, P. Le Fèvre, E. Lhuillier, T. Wakamura, C. Mattevi, J. E. Rault, M. Calandra, and A. Ouerghi, Electronic band structure of Two-Dimensional WS2/graphene van der Waals Heterostructures, Physical Review B 97, 155421 (2018).
  * Scalise  _et al._ [2012] E. Scalise, M. Houssa, G. Pourtois, V. Afanas’ev, and A. Stesmans, Strain-induced semiconductor to metal transition in the two-dimensional honeycomb structure of MoS2, Nano Research 5, 43 (2012).
  * Yun  _et al._ [2012] W. S. Yun, S. W. Han, S. C. Hong, I. G. Kim, and J. D. Lee, Thickness and strain effects on electronic structures of transition metal dichalcogenides: 2H-Mx2 semiconductors (M=Mo, W; X= S, Se, Te), Physical Review B 85, 033305 (2012).
  * Ebnonnasir  _et al._ [2014] A. Ebnonnasir, B. Narayanan, S. Kodambaka, and C. V. Ciobanu, Tunable MoS2 bandgap in MoS2-graphene heterostructures, Applied Physics Letters 105, 031603 (2014).
  * Junior  _et al._ [2023] P. E. F. Junior, T. Naimer, K. M. McCreary, B. T. Jonker, J. J. Finley, S. A. Crooker, J. Fabian, and A. V. Stier, Proximity-enhanced valley Zeeman splitting at the WS2/graphene interface, 2D Materials 10, 034002 (2023).
  * Magnozzi  _et al._ [2020] M. Magnozzi, M. Ferrera, G. Piccinini, S. Pace, S. Forti, F. Fabbri, C. Coletti, F. Bisio, and M. Canepa, Optical dielectric function of two-dimensional WS2 on epitaxial graphene, 2D Materials 7, 025024 (2020).
  * Riis-Jensen  _et al._ [2020] A. C. Riis-Jensen, J. Lu, and K. S. Thygesen, Electrically controlled dielectric band gap engineering in a two-dimensional semiconductor, Physical Review B 101, 121110 (2020).
  * Tebbe  _et al._ [2023] D. Tebbe, M. Schütte, K. Watanabe, T. Taniguchi, C. Stampfer, B. Beschoten, and L. Waldecker, Tailoring the dielectric screening in WS2–graphene heterostructures, npj 2D Materials and Applications 7, 1 (2023).
  * Latini  _et al._ [2015] S. Latini, T. Olsen, and K. S. Thygesen, Excitons in van der Waals heterostructures: The important role of dielectric screening, Physical Review B 92, 245123 (2015).
  * Karni  _et al._ [2022] O. Karni, E. Barré, V. Pareek, J. D. Georgaras, M. K. Man, C. Sahoo, D. R. Bacon, X. Zhu, H. B. Ribeiro, A. L. O’Beirne, _et al._ , Structure of the moiré exciton captured by imaging its electron and hole, Nature 603, 247 (2022).
  * Naik  _et al._ [2022] M. H. Naik, E. C. Regan, Z. Zhang, Y.-H. Chan, Z. Li, D. Wang, Y. Yoon, C. S. Ong, W. Zhao, S. Zhao, M. I. B. Utama, B. Gao, X. Wei, M. Sayyad, K. Yumigeta, K. Watanabe, T. Taniguchi, S. Tongay, F. H. da Jornada, F. Wang, and S. G. Louie, Intralayer charge-transfer moiré excitons in van der Waals superlattices, Nature 609, 52 (2022).
  * Barré  _et al._ [2022] E. Barré, O. Karni, E. Liu, A. L. O’Beirne, X. Chen, H. B. Ribeiro, L. Yu, B. Kim, K. Watanabe, T. Taniguchi, K. Barmak, C. H. Lui, S. Refaely-Abramson, F. H. da Jornada, and T. F. Heinz, Optical absorption of interlayer excitons in transition-metal dichalcogenide heterostructures, Science 376, 406 (2022).
  * Kundu  _et al._ [2023] S. Kundu, T. Amit, H. Krishnamurthy, M. Jain, and S. Refaely-Abramson, Exciton fine structure in twisted transition metal dichalcogenide heterostructures, npj Computational Materials 9, 186 (2023).
  * Kundu  _et al._ [2022] S. Kundu, M. H. Naik, H. Krishnamurthy, and M. Jain, Moiré induced topology and flat bands in twisted bilayer wse 2: A first-principles study, Physical Review B 105, L081108 (2022).
  * Hernangómez-Pérez  _et al._ [2023] D. Hernangómez-Pérez, A. Kleiner, and S. Refaely-Abramson, Reduced Absorption Due to Defect-Localized Interlayer Excitons in Transition-Metal Dichalcogenide–Graphene Heterostructures, Nano Letters 23, 5995 (2023).
  * Refaely-Abramson  _et al._ [2018] S. Refaely-Abramson, D. Y. Qiu, S. G. Louie, and J. B. Neaton, Defect-Induced Modification of Low-Lying Excitons and Valley Selectivity in Monolayer Transition Metal Dichalcogenides, Physical Review Letters 121, 167402 (2018).
  * Amit  _et al._ [2022] T. Amit, D. Hernangómez-Pérez, G. Cohen, D. Y. Qiu, and S. Refaely-Abramson, Tunable magneto-optical properties in mos2 via defect-induced exciton transitions, Physical Review B 106, L161407 (2022).
  * Kohn and Sham [1965] W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Physical Review 140, A1133 (1965).
  * Hybertsen and Louie [1986] M. S. Hybertsen and S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies, Physical Review B 34, 5390 (1986).
  * Raja  _et al._ [2017] A. Raja, A. Chaves, J. Yu, G. Arefe, H. M. Hill, A. F. Rigosi, T. C. Berkelbach, P. Nagler, C. Schüller, T. Korn, C. Nuckolls, J. Hone, L. E. Brus, T. F. Heinz, D. R. Reichman, and A. Chernikov, Coulomb engineering of the bandgap and excitons in two-dimensional materials, Nature Communications 8, 15251 (2017).
  * Waldecker  _et al._ [2019] L. Waldecker, A. Raja, M. Rösner, C. Steinke, A. Bostwick, R. J. Koch, C. Jozwiak, T. Taniguchi, K. Watanabe, E. Rotenberg, T. O. Wehling, and T. F. Heinz, Rigid Band Shifts in Two-Dimensional Semiconductors through External Dielectric Screening, Physical Review Letters 123, 206403 (2019).
  * Yuan  _et al._ [2018] L. Yuan, T.-F. Chung, A. Kuc, Y. Wan, Y. Xu, Y. P. Chen, T. Heine, and L. Huang, Photocarrier generation from interlayer charge-transfer transitions in WS2-graphene heterostructures, Science Advances 4, e1700324 (2018).
  * Giusca  _et al._ [2019] C. E. Giusca, Z. Lin, M. Terrones, D. K. Gaskill, R. L. Myers-Ward, and O. Kazakova, Probing exciton species in atomically thin WS2–graphene heterostructures, Journal of Physics: Materials 2, 025001 (2019).
  * Coy Diaz  _et al._ [2015] H. Coy Diaz, J. Avila, C. Chen, R. Addou, M. C. Asensio, and M. Batzill, Direct Observation of Interlayer Hybridization and Dirac Relativistic Carriers in Graphene/MoS2 van der Waals Heterostructures, Nano Letters 15, 1135 (2015).
  * Rohlfing and Louie [2000] M. Rohlfing and S. G. Louie, Electron-hole excitations and optical spectra from first principles, Physical Review B 62, 4927 (2000).
  * Rösner  _et al._ [2016] M. Rösner, C. Steinke, M. Lorke, C. Gies, F. Jahnke, and T. O. Wehling, Two-Dimensional Heterojunctions from Nonlocal Manipulations of the Interactions, Nano Letters 16, 2322 (2016).
  * Stauber  _et al._ [2008] T. Stauber, N. M. R. Peres, and A. K. Geim, Optical conductivity of graphene in the visible region of the spectrum, Physical Review B 78, 085432 (2008).
  * Nair  _et al._ [2008] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, Fine Structure Constant Defines Visual Transparency of Graphene, Science 320, 1308 (2008).
  * Lorchat  _et al._ [2020] E. Lorchat, L. E. P. López, C. Robert, D. Lagarde, G. Froehlicher, T. Taniguchi, K. Watanabe, X. Marie, and S. Berciaud, Filtering the photoluminescence spectra of atomically thin semiconductors with graphene, Nature Nanotechnology 15, 283 (2020).
  * Deslippe  _et al._ [2012] J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures, Computer Physics Communications 183, 1269 (2012).
  * Fang  _et al._ [2022] J. Fang, K. Yao, T. Zhang, M. Wang, T. Jiang, S. Huang, B. A. Korgel, M. Terrones, A. Alù, and Y. Zheng, Room-Temperature Observation of Near-Intrinsic Exciton Linewidth in Monolayer WS2, Advanced Materials 34, 2108721 (2022).
  * Selig  _et al._ [2016] M. Selig, G. Berghäuser, A. Raja, P. Nagler, C. Schüller, T. F. Heinz, T. Korn, A. Chernikov, E. Malic, and A. Knorr, Excitonic linewidth and coherence lifetime in monolayer transition metal dichalcogenides, Nature Communications 7, 13279 (2016).
  * Chan  _et al._ [2023] Y.-h. Chan, J. B. Haber, M. H. Naik, J. B. Neaton, D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Exciton Lifetime and Optical Line Width Profile via Exciton–Phonon Interactions: Theory and First-Principles Calculations for Monolayer MoS2, Nano Letters 23, 3971 (2023).
  * Olivero and Longbothum [1977] J. J. Olivero and R. L. Longbothum, Empirical fits to the Voigt line width: A brief review, Journal of Quantitative Spectroscopy and Radiative Transfer 17, 233 (1977).