# Optoelectronic properties of the CuI, AgI and Janus Cu2BrI, and Ag2BrI
monolayers by many-body perturbation theory

Mohammad Ali Mohebpour<EMAIL_ADDRESS>Department of Physics,
University of Guilan, P. O. Box 41335-1914, Rasht, Iran  Bohayra Mortazavi
<EMAIL_ADDRESS>Department of Mathematics and Physics, Leibniz
Universität Hannover, Appelstra$\beta$e 11,30167 Hannover, Germany  Xiaoying
Zhuang  Department of Mathematics and Physics, Leibniz Universität Hannover,
Appelstra$\beta$e 11,30167 Hannover, Germany  Meysam Bagheri Tagani
<EMAIL_ADDRESS>Department of Physics, University of Guilan, P. O. Box
41335-1914, Rasht, Iran

###### Abstract

In an outstanding experimental advance in the field of two-dimensional
nanomaterials, cuprous iodide (CuI) and silver iodide (AgI) monolayers have
been grown via a novel graphene encapsulation synthesis approach
[Adv.Mater.2022, 34, 2106922]. Inspired by this accomplishment, we conduct
first-principles calculations to investigate the elastic, phononic thermal
transport, electronic, and optical properties of the native CuI and AgI and
Janus Cu2BrI and Ag2BrI monolayers. Electronic and excitonic optical
properties are elaborately studied using the many-body perturbation theory on
the basis of GW approximation. Our results indicate that these novel systems
are stable but with soft elastic modulus and ultralow lattice thermal
conductivity. It is also shown that the studied monolayers are wide-gap
semiconductors with exciton binding energies close to 1 eV. The effects of
mechanical straining and electric field on the resulting electronic and
optical properties are also analyzed. The presented first-principles results
provide a deep understanding of the stability, phononic transport and tunable
optoelectronic properties of the native CuI and AgI and Janus Cu2BrI and
Ag2BrI monolayers, which can serve as a guide for the oncoming studies.

CuI; Two-dimensional materials; Electronic properties; Excitonic properties;
Thermal conductivity.

##  I Introduction

Two-dimensional (2D) materials have attracted the interest of researchers in
recent years, for developing next-generation optoelectronic and energy
storage/conversion devices due to their outstanding physical properties.
Graphene, the most famous 2D material, has been proven to strongly interact
with light from microwave to ultraviolet wavelengths, exhibiting wide
applications in optoelectronic devices [1]. To improve the efficiency of
optoelectronic devices, novel layered 2D materials have been experimentally
synthesized such as transition metal dichalcogenide (TMD) monolayers (like
MoS2 [2], WS2 [3], and WSe2 [4]), metal oxides [5, 6], metal perovskite [7],
and MXenes [8, 9]. All aforementioned monolayers are exfoliated from their
bulk lattices due to the layered structures, and they usually exhibit
unconventional physical properties, distinct from their bulk counterparts.

The majority of 2D nanomaterials fabricated to date have layered structures,
in which there exist weak van der Waals interlayer interaction. These weak
interactions facilitate the exfoliation process and enable the possibility to
derive single or few layers. The synthesis of non-layered 2D materials,
despite many challenges, not only significantly expands the field of 2D
materials, but also opens new horizons in materials science. One of the first
synthesized monoelemental non-layered 2D materials was boron allotropes,
borophene [10, 11], which revolutionized the study of rechargeable batteries
[12] and hydrogen storage [13]. The non-layered monochalcogenides of ZnS [14]
and PbS [15] were also synthesized by bottom-up approach, exhibiting unique
features for applications in solar water splitting and field-effect
transistor, respectively.

Cuprous iodide, CuI, as a group 11 transition metal halide is a non-layered
material and crystallizes in different phases [16]. At temperature below 370
0C , it is crystallized in cubic $\gamma$-phase. In the temperature range of
370-407 0C , it is in hexagonal $\beta$-phase and a transition to cubic
$\alpha$-phase is happened for higher temperatures. The bulk $\gamma$-phase of
CuI is known for a high thermopower coefficient and substantial optoeletric
properties[17]. Very recently, ultrathin nanosheets of $\gamma$-phase CuI has
been synthesized on SiO2/Si substrate using a facile physical vapor deposition
process [18]. It was also shown that the CuI nanosheets can be synthesized on
the 2D substrates like WS2 and WSe2. Mustonen et al. [19] successfully
synthesized monolayer CuI at ambient conditions by a single-step wet-chemical
process between graphene layers. However, the mentioned articles did not
provide any information on the electronic and optical properties of CuI.
Indeed, no optical analysis was performed to determine the value of band gap
and its type. In addition, the effect of dimensional reduction on the change
of electronic properties of the monolayer in comparison with its bulk
counterpart was not discussed.

Motivated by the recent advance on the synthesis of monolayer CuI via a novel
graphene encapsulation synthesis approach, we herein investigate the
electronic, excitonic, optical, and thermal properties of monolayer group 11
transition metal halides, including native CuI and AgI and Janus counterparts
of Cu2IBr and Ag2IBr using first-principles calculations. The electronic and
optical properties of the considered monolayers are studied using many-body
perturbation calculations, and the excitonic optical properties are calculated
by solving the Bethe-Salpeter equation. The dynamical and thermal stabilities
of the monolayers are found to be validated by phonon dispersion spectra and
ab-initio molecular dynamics simulations, respectively. The quasiparticle band
gaps are obtained to be 4.42, 4.26, 3.93, and 3.96 eV for CuI, AgI, Cu2BrI,
and Ag2BrI monolayers, respectively. We also examine the effects of strain and
external electric field on the spin-orbit induced band splitting and find an
increase of more than 100$\%$ in the magnitude of band splitting for the
conduction band of Ag2IBr under a uniaxial strain of 3$\%$. In addition, the
electric field produces a noticeable band splitting in the non-Janus
monolayers. The first bright excitons of CuI, AgI, Cu2BrI, and Ag2BrI
monolayers are found to be tightly bound with binding energies of 0.88, 0.92,
0.95, and 0.99 eV, respectively, meaning the super stability of the excitonic
states against thermal decomposition at 300 K. To obtain the phonon thermal
conductivity of the monolayers, the moment tensor potentials are trained over
ab-initio molecular dynamics datasets. Then, the lattice thermal transport is
evaluated using the full-iterative solution of the Boltzmann transport
equation. The room-temperature thermal conductivity of CuI, AgI, Cu2BrI, and
Ag2BrI monolayers are predicted to be remarkably low, 3.75, 2.27, 3.13, and
1.26 W/mK, respectively. The presented first-principles results provide
important understanding about the key physical properties of the non-Janus CuI
and AgI and Janus Cu2BrI and Ag2BrI monolayers, which can be useful for their
practical application and can guide oncoming theoretical and experimental
works.

##  II Computational methods

The density functional theory (DFT) calculations are performed using the
Vienna Ab-initio Simulation Package (VASP) [20] on the basis of the
generalized gradient approximation (GGA) [21] and the Perdew-Burke-Ernzerhof
functional for solids (PBEsol) [22]. The plane-wave and self-consistent loop
energy cutoffs are set to 600 eV and 10-7 eV, respectively. The optimized
lattice structures are obtained using the conjugate gradient algorithm until
the Hellman-Feynman forces drop below 10-3 eV/Å, considering a fixed vacuum
space of 20 Å along the thickness of the monolayers. The first Brillouin zone
is sampled with a 15$\times$15$\times$1 Monkhorst-Pack k-point mesh [23] based
on the convergence of the total energy within 10-6 eV. The electronic band
structure is also evaluated using the Heyd-Scuseria-Ernzerhof (HSE06) hybrid
functional with the default mixing parameter ($\alpha=0.25$) [24]. The charge
transfer between atoms is determined using the Bader analysis [25].

The first-principles many-body perturbative calculations are carried out using
the single-shot GW method [26], referred to as G0W0. In these calculations,
the quasiparticle (QP) band gap should be carefully converged with respect to
the number of virtual bands, number of frequency grid points, kinetic energy
cutoff, and k-point mesh. After the convergence test, a set of 168, 204, 152,
and 186 virtual bands is employed in the G0W0 calculations for CuI, AgI,
Cu2BrI, and Ag2BrI monolayers, respectively. The number of frequency grid
points is selected to be 96. The kinetic energy cutoff for the plane wave and
the response function is set to be 600 and 200 eV, respectively. The Brillouin
zone is integrated with a 9$\times$9$\times$1 $\Gamma$-centered k-point mesh.
The QP band structure of the G0W0 calculations is interpolated using the
maximally localized Wannier functions (MLWFs), as implemented in the WANNIER90
code [27]. Here, the number of Wannier bands is selected to be 24, and the
$sp^{3}d^{2}$ hybrid orbitals are chosen for the initial projections. The
excitonic optical properties are studied by calculating the macroscopic
dielectric function, given as
$\varepsilon(\omega)=\varepsilon_{1}(\omega)+i\varepsilon_{2}(\omega)$,
through solving the Bethe-Salpeter equation (BSE) [28] on top of the G0W0
eigenvalues. Here, the Tamm-Dancoff approximation (TDA) is used, which
excludes the resonant-antiresonant coupling [29]. The imaginary part of the
dielectric function is obtained by summation over the empty conduction bands,
and the real part is transformed by the Kramers-Kronig relation, as
elaborately discussed in our previous study [30]. To check the effects of
self-consistency in the excitonic properties, we perform the fully self-
consistent GW (i.e. QPGW). At this level, the QP energies and one-electron
orbitals are updated 4 times in the calculations of the Green’s function ($G$)
and screened Coulomb interaction ($W$). The 15 highest occupied valence bands
and the 15 lowest unoccupied conduction bands are taken into account to get a
converged BSE spectrum. For optical calculations, the spin-orbit coupling
(SOC) is considered.

To evaluate the phononic properties, the moment tensor potentials (MTPs) [31]
are developed using the MLIP package [32]. The ab-initio molecular dynamics
(AIMD) simulations are carried out with a time step of 1 fs and a Monkhorst-
Pack k-point mesh of 2$\times$2$\times$1 over 4$\times$3$\times$1 supercells.
Two separate AIMD calculations are conducted with the controlled temperature
from 2 to 100 K and from 300 to 1000 K, each for 1000 time steps. Complete
AIMD trajectories are then subsampled, and around 800 configurations are
selected to train the MTPs. The phonon dispersion relations and harmonic 2nd
order interatomic force constants are obtained using the PHONOPY code [33]
over 6$\times$6$\times$1 supercells [34]. The anharmonic 3rd order interatomic
force constants are calculated using the MTPs over 6$\times$6$\times$1
supercells [35], considering the interactions with eleventh nearest neighbors.
The lattice thermal transport is evaluated using the full-iterative solution
of the Boltzmann transport equation (BTE) via employing the ShengBTE package
[36], considering the isotope scattering.

##  III Results and discussion

###  III.1 Structural properties

After the experimental achievement by Mustonen and coworkers [19], we found
CuI and AgI monolayers very interesting for exploring their physical
properties. These systems belong to the group 11 transition-metal halide MX (M
= Cu, Ag, Au; X = Cl, Br, I) 2D lattices, among which MCl and AuX monolayers
are thermally and dynamically unstable [37]. We also constructed two Janus
monolayers based on CuI and AgI native monolayers to seek fascinating
properties caused by inversion symmetry breaking. For this purpose, we
replaced the I atoms on one side of the native monolayers with Br atoms,
because with Cl atoms the stability is expected to be low. By and large, in
this work, we study non-Janus CuI and AgI and Janus Cu2BrI and Ag2BrI
monolayers, which all have highly symmetrical hexagonal lattice. Fig. 1
illustrates the top and side views of the fully-relaxed CuI monolayer as a
representative. Obviously, the non-Janus monolayers show inversion and mirror
symmetries, however, the Janus ones have only mirror symmetry. That is to say,
the inversion symmetry is broken in the Janus monolayers due to the
replacement of the I atom by Br atom. The relaxed lattice constant of CuI,
AgI, Cu2BrI, and Ag2BrI monolayers according to the PBEsol functional is
predicted to be 4.059, 4.402, 3.962, and 4.326 Å, respectively. Based on
experimental tests [19], the distances between the Cu atomic planes and the
Cu-I bond lengths along the in-plane and out-of-plane directions are
1.63$\pm$0.35, 2.67$\pm$0.16, and 2.55$\pm$0.49 Å, respectively, which are in
good agreement with the corresponding values of 1.345, 2.642 and 2.566 Å,
respectively.

Figure 1: Top and side views of the relaxed CuI monolayer with the first
Brillouin zone. The orange dashed lines show the unit cell. The orange arrows
show the mirror and inversion symmetries. The in-plane and out-of-plane Cu-I
bond lengths are given.

We firstly study the stability of CuI, AgI, Cu2BrI, and Ag2BrI monolayers on
the basis of the phonon dispersion relations. The obtained phonon dispersion
spectra along the high symmetry points of the first Brillouin zone are
depicted in Fig. S1 of the supporting information document. Apparently, there
is no imaginary frequency in the phonon spectra, confirming their dynamical
stability. The thermal stability of monolayers is then examined by the AIMD
simulations conducted at 500 K. As shown in Fig. S2, during the simulations,
the total energy of the monolayers fluctuates around the mean values,
confirming their thermal stability up to 500 K. Interestingly, the $C_{11}$
($C_{12}$) elastic constant of CuI, AgI, Cu2BrI, and Ag2BrI monolayers is
calculated to be 47 (21), 26 (15), 44 (21), and 24 (15) N/m, which reveal that
these novel 2D systems are soft and include weak bonding interactions.
Moreover, it is found that the systems with Cu atoms show distinctly higher
rigidity than the counterparts based on Ag atoms. According to the
aforementioned findings, the CuI, AgI, Cu2BrI, and Ag2BrI monolayers are very
stable but generally among soft nanomaterials.

###  III.2 Electronic and optical properties

After assuring the stability of the monolayers, we turn our attention to their
electronic properties. Fig. 2 displays the band structures of the monolayers
together with their total and partial density of states (DOS). At the PBE
(HSE06) level, the monolayers are semiconductors with direct band gaps of 2.10
(3.33), 1.99 (3.04), 1.53 (2.84), and 1.61 (2.71) eV for CuI, AgI, Cu2BrI, and
Ag2BrI, respectively. Except for the band gaps, the characteristics of the
band structures are the same at both levels. That is to say, including a
fraction (0.25) of the exact Hartree-Fock exchange energy in the DFT
formulation only corrects the band gaps by 1.23, 1.05, 1.31, and 1.10 eV for
CuI, AgI, Cu2BrI, and Ag2BrI, respectively. For all the monolayers, the
valence band maximum (VBM) and the conduction band minimum (CBM) are located
at the $\Gamma$ point. The conduction band edges are quadratically dispersed,
that show free electrons. The valence band edges are nearly flat, that
represent localized holes. Although the valence bands edges at the $\Gamma$
point show double degeneracy, there is a tiny gap of $10^{-3}$ eV between the
heavy and light hole subbands. It is worth noting that the band gaps
calculated for CuI and AgI monolayers agree very well with those reported in a
previous study [37].

From the DOS, it is seen that the CBMs are equally contributed by the
$s$-orbitals of the Cu/Ag atoms and the $p$-orbitals of the Br/I atoms while
the VBMs are mostly controlled by the $d$-orbitals of the Cu/Ag atoms and the
$p$-orbitals of the I atoms. Therefore, the optical selection rules allow the
transitions between the $p$-orbitals of the halogen atoms and the $s$-orbitals
of the transition metal ones. No considerable participation from Br atoms is
found near the edge of valence band while contributing the most at the edge of
conduction band in the Janus monolayers, which is in line with the reduction
of the band gap by substituting Br atoms. The shapes of the wave functions at
the VBMs and CBMs support the above discussion. As shown in the insets of Fig.
2, at the VBMs of the non-Janus monolayers, the wave functions are shaped like
dumbbells, which are distributed along the x-direction and centered on the I
atoms, showing the $p_{x}$-orbitals of the I atoms. At the CBMs, the wave
functions are formed spherically on the Cu/Ag atoms and hemispherically on the
I atoms, referring to as $s$ and $p_{z}$ orbitals, respectively. For Ag2BrI
and Cu2BrI Janus monolayers, the wave functions of the VBMs exhibit a
hybridization between the $p_{x}$ and $p_{y}$-orbitals of the I atoms. At the
CBMs, one can see the predominant contribution of the $p_{z}$-orbitals of the
halogen atoms.

Figure 2: Electronic band structures of the monolayers at the PBE (blue lines)
and HSE06 (red lines) levels together with the density of states at the PBE
level. The VBMs are set to zero. The squares of the wave functions at the VBM
and CBM are shown in the insets. Figure 3: Charge density difference of the
monolayers with the Bader charge transfer between atoms. The isosurface of
charge accumulation and depletion was set to 5$\times$10-3 e/bohr3.

The charge density difference (CDD) isosurface of the monolayers is depicted
in Fig. 3, where the yellow and cyan colors denote the charge accumulation and
depletion, respectively. Apparently, the highest charge accumulation is
located at the center of the atomic bonds, indicating the covalent bonding
characteristics. The highest charge depletion is around the transition metal
atoms. In accordance with the CDD analysis, for CuI structure, we find that
0.273 $e$ is transferred from the Cu to I atoms. While, for AgI structure,
0.237 $e$ is transported from the Ag to I atoms. These are in line with the
larger electronegativity of the I (2.66) atom compared to the Cu (1.90) and Ag
(1.93) transition metals. In Cu2BrI Janus monolayer, the charge transfer from
the two neighboring Cu atoms to the uppermost and lowermost layers of the
halogen atoms is different. That is to say, the Br atoms receive more charge
compared to the I atoms due to the larger electronegativity of Br (2.96) atom.
Also, in Ag2BrI Janus monolayer, the two neighboring Ag atoms lose different
amounts of charge in such a manner that 65% of the total transferred charge is
delivered to the Br atoms and the rest (0.213 $e$) is sent to the I atoms.
These differences result in a potential gradient normal to the basal plane,
forming an intrinsic internal electric field between the uppermost and
lowermost layers. The intrinsic out-of-plane electric field caused by
inversion symmetry breaking in the Janus monolayers might lead to spin-valley
splitting and Rashba spin splitting when combined with the SOC.

Figure 4: (a) Spin-orbit induced valence and conduction bands splitting and
(b) band gap variation of Cu2BrI and Ag2BrI Janus monolayers as a function of
uniaxial strain along the armchair and zigzag directions. (c) Spin-orbit
induced valence band splitting of AgI monolayer under different electric
fields. The rest of the results are available in Fig. S5.(d) Band gap
variation of the monolayers as a function of electric field.

We check out the effects of the SOC in the band structures. By including the
SOC at the PBE (HSE06) level, the numbers of energy bands are doubled and the
band gaps are reduced to 1.86 (3.03), 1.72 (2.73), 1.31 (2.58), and 1.37
(2.44) eV for CuI, AgI, Cu2BrI, and Ag2BrI monolayers, respectively. As shown
in Fig. S3 (Fig. S4), for CuI and AgI monolayers, the SOC cannot remove the
double degeneracy of the individual energy states while for the Janus
monolayers, the bands are split. This spin splitting, referred to as Zeeman-
type splitting, is in fact a consequence of the SOC combined with inversion
symmetry breaking. At the PBE level, the spin splitting constant at the
$\Gamma$ point is zero but at the $K$ point, the spin splitting of valence
(conduction) band edge is 12 (28) and 13 (45) meV for Cu2BrI and Ag2BrI
monolayers, respectively. The SOC also increases the tiny gap observed between
the heavy and light hole subbands at the $\Gamma$ point up to 400 and 450 meV
for the non-Janus and Janus monolayers, respectively. At the HSE06 level, the
strength of SOC is predicted to be higher. Hence, the reported values for spin
splitting will be slightly larger at this level. By and large, one can
conclude that the main effect of the SOC is on the valence bands owing to the
hybridization between the $d$-orbitals of the Cu/Ag atoms and the $p$-orbitals
of the I atoms. The conduction bands are not influenced by the SOC due to the
predominant participation of the $s$-orbitals of the Cu/Ag atoms.

In experimental syntheses, 2D materials are grown on different kinds of metal
or insulator substrates. These substrates can affect the intrinsic properties
of materials. The most common effect of the substrate is applying strain to
the overlayer. If the substrate is active, the interactions between the
substrate and 2D material can strongly affect the electronic properties of the
overlayer. In this part, we examine the dependence of spin-orbit induced band
splitting on the applied strain. It should be noted that strain is only
applied in the form of uniaxial in the armchair and zigzag directions because
it can break the inversion symmetry of materials. The combination of spin-
orbit with inversion symmetry breaking produces attractive effects such as
Rashba [38] and valleytronic [39]. Inversion symmetry is inherently broken in
Janus structures, and as noted, the SOC causes band splitting at the $K$
points. The maximum band splitting in Cu2BrI and Ag2BrI Janus monolayers as a
function of strain is plotted in Fig. 4 (a). Irrespective of direction, strain
slightly reduces the valence band splitting. In contrast, we see an increase
in the conduction band splitting with strain. The stronger effect of spin-
orbit interaction in the conduction band is due to the predominant role of the
transition metal in the conduction band, which is observable from the DOS. The
rate of increase for strain in the armchair direction is slightly higher than
that in the zigzag direction. In addition, the increase in Ag2BrI structure is
much more intense than in Cu2BrI monolayer. A strain of about 1$-$3% in Ag2BrI
monolayer can cause an increase of more than 100% in the magnitude of band
splitting for the conduction band. Strong spin-orbit interactions, inherent
electric field, and increased inversion symmetry breaking by strain are the
main reasons for the observed increase.

We additionally examine the spin-orbit induced band splitting of CuI and AgI
monolayers in terms of the applied uniaxial strain. Interestingly, although
strain can slightly alter the symmetries of the structures, no band splitting
induced by spin-orbit interaction is observed in these monolayers (Data not
shown). The softness of the materials, which was also confirmed in
experimental reports [19], is a major obstacle for engineering the band
structures and creating the Zeeman-type or Rashba-type spin splitting with
strain. Our evaluation shows that applying strain is not a good approach for
tuning the properties of CuI and AgI monolayers. Hence, they can be grown on
substrates with different lattice constants while showing the same electronic
properties.

Fig. 4 (b) illustrates the variations of the band gaps of Cu2BrI and Ag2BrI
Janus monolayers as a function of strain. Apparently, an increase of 20% (10%)
is observed in the band gap of Ag2BrI monolayer under the strain of 5% in the
armchair (zigzag) direction. This is attributed to the presence of a heavy
transition metal such as silver and the absence of an inversion symmetry
center in this Janus monolayer, which are critical for the noticeable change
in the band gap. The variation of the band gap in Ag2BrI monolayer is much
more intense than that in Cu2BrI, which is in line with the smaller elastic
constants of Ag2BrI monolayer. In other structures, the changes in band gaps
under strain up to 5% are less than 3% (Data not shown).

We also explore the spin-orbit induced band splitting in the presence of an
external electric field. In contrast to strain, the electric field produces a
noticeable band splitting in CuI and AgI monolayers. The splitting is visible
throughout the Brillouin zone, as shown in Fig. 4 (c) and Fig. S5. For AgI
(CuI) monolayer, the largest valence band splitting occurs in the $K-M$ path
and can reach 140 (73) meV under an electric field of 0.5 eV/Å, which is
comparable to transition metal dichalcogenide monolayers. The largest
conduction band splitting occurs at the $K$ point. Also, a Rashba-type band
splitting is observed in the $\Gamma-K$ path under strong electric fields.
Comparing the band splitting in CuI and AgI monolayers shows that it is
directly related to the transition metal atoms. The larger the atomic number
is, the greater the band splitting will be. However, in the Janus structures,
the dependence of band splitting on the electric field is very weak as shown
in Fig. S5. Due to the lack of an inversion center, there is an inherent
electric field in these structures that leads to a noticeable band splitting
as discussed above. Therefore, the external electric field cannot
significantly change the band splitting. The interesting point is that, unlike
the non-Janus structures, increasing the strength of the external field has no
effect on the magnitude of band splitting. In addition, the direction of the
electric field does not have a significant effect on the results.

The variation of the band gaps as a function of external electric field is
shown in Fig. 4 (d). In CuI and AgI non-Janus monolayers, due to the inversion
symmetry and the arrangement of atoms in the lower and upper planes, the
changes in the band gaps are independent of the direction of electric field.
By increasing the intensity of electric field, the band gaps decrease and the
reduction is more for CuI monolayer. On the contrary, in the Janus structures,
the changes in the band gaps in addition to the magnitude of electric field
depend on its direction. A linear increase is observed in the band gaps with
the electric field in the -z direction. While we see a decrease in the band
gaps by increasing the electric field in the +z direction. The band gaps
reduction directly depends on the difference in charge transferred between two
halogen atoms. As the electric field increases in the +z direction, we observe
more charge transfer towards the Br atoms, which reduces the band gaps. In
contrast, the external electric field in the -z direction is in the opposite
direction of the intrinsic electric field caused by the dipole inside the
monolayers, which reduces the difference in the charge accumulated on halogen
atoms, giving rise to the band gaps.

To capture the role of electron-electron exchange interaction in the
electronic structure, we step beyond the PBE and HSE06 levels of theory using
the non-self-consistent GW approach, known as G0W0. The G0W0 band structures
of the monolayers in the absence of the SOC interaction are available in Fig.
S6. As can be noticed, the general shapes of the band structures are similar
to those calculated with the PBE functional provided in Fig. 2. All the
monolayers still have direct band gaps at the $\Gamma$ point. However, the
values of band gaps are different from the PBE calculations due to the self-
energy corrections. At this level, the QP band gaps are obtained to be 4.42,
4.26, 3.93, and 3.96 eV for CuI, AgI, Cu2BrI, and Ag2BrI monolayers,
respectively. Such large self-energy corrections of 2.32, 2.27, 2.40, and 2.35
eV emphasize the importance of the electron-electron interaction in these
structures. Also, it is found that increasing the atomic weight of the halogen
(transition metal) atom increases (decreases) the QP band gap of the
monolayer. Owing to the slight deformation in the bands dispersion, a rigid
shift of the conduction bands with respect to the VBMs does not describe the
self-energy effects properly. Including the SOC reduces the QP band gaps down
to 4.02, 3.85, 3.59, and 3.61 eV for CuI, AgI, Cu2BrI, and Ag2BrI monolayers,
respectively. As expected, the structure with heavier atoms has the strongest
SOC. To better compare, we listed the band gaps calculated at different levels
in Table 1.

Table 1: Band gaps calculated for the considered monolayers at different levels of theory in the unit of eV. Structure | PBE | PBE$+$SOC | HSE06 | HSE06$+$SOC | G0W0 | G0W${}_{0}+$SOC | Optical
---|---|---|---|---|---|---|---
CuI | 2.10 | 1.86 | 3.33 | 3.03 | 4.42 | 4.02 | 3.14
AgI | 1.99 | 1.72 | 3.04 | 2.73 | 4.26 | 3.85 | 2.93
Cu2BrI | 1.53 | 1.31 | 2.84 | 2.58 | 3.93 | 3.59 | 2.64
Ag2BrI | 1.61 | 1.37 | 2.71 | 2.44 | 3.96 | 3.61 | 2.62
Figure 5: Imaginary part of the dielectric function of the monolayers at
different levels of theory including the DFT$+$RPA (electron$-$electron and
electron$-$hole interactions excluded), G0W${}_{0}+$RPA (electron$-$electron
interaction included and electron$-$hole interaction excluded), and
G0W${}_{0}+$BSE (electron$-$electron and electron$-$hole interactions
included) together with the optical oscillator strength. The exciton binding
energy ($E_{b}$), the difference between the G0W${}_{0}+$SOC band gap and the
optical gap, is given for each monolayer. Figure 6: Amplitude of transition
dipole moment of the monolayers along the high symmetry points of the
Brillouin zone.

We study the excitonic optical properties of the monolayers by solving the
Bethe-Salpeter equation over the G0W0 eigenvalues (i.e., G0W${}_{0}+$BSE),
which includes the electron-electron and electron-hole interactions. These
interactions have been shown to play a vital role in the optical properties of
2D materials [40, 41]. To clarify the role of the many-body effects, we also
calculated the optical spectra of the monolayers using the random-phase
approximation (RPA) over the G0W0 (i.e., G0W${}_{0}+$RPA) and the DFT (i.e.,
DFT$+$RPA), where the aforementioned interactions are excluded. Owing to the
symmetric and isotropic crystal structure of the monolayers, the optical
coefficients along the x-direction ($E\parallel x$) are identical to those
along the y-direction ($E\parallel y$). On the other hand, due to the strong
depolarization effect in the low-dimensional systems for light polarization
perpendicular to the plane ($E\parallel z$), the optical coefficients along
the z-direction are negligible. Therefore, we only study the optical response
of the monolayers for the in-plane x-polarized light. Fig. 5 illustrates the
imaginary part of the dielectric function of the monolayers at three different
levels of theory for light polarized along the x-direction. As it is obvious,
at the DFT$+$RPA level, the optical spectra of the monolayers show numerous
peaks, which are associated with the doubling of bands in the presence of the
SOC interaction. At this level, the first peak appears at 1.85, 1.71, 1.30,
and 1.36 eV for CuI, AgI, Cu2BrI, and Ag2BrI monolayers, respectively. These
peaks correspond to direct transitions from the highest valence band to the
lowest conduction band at the $\Gamma$ point. Including the electron-electron
interaction (i.e., G0W${}_{0}+$RPA) results in a blue shift in the optical
spectra and reduces the intensity of the peaks overestimated by the DFT$+$RPA
level. The electron-electron interaction also decreases the number of peaks.
On the other hand, considering the electron-hole interaction leads to a
cancelation effect and subsequently a redshift in the optical spectra. The
electron-hole interaction modifies the general shape of the spectra. At this
level (i.e., G0W${}_{0}+$BSE), the first peaks, referred to as optical gaps,
appear at 3.14, 2.93, 2.64, and 2.62 eV, which correspond to tightly bound
excitons with binding energies of 0.88, 0.92, 0.95, and 0.99 eV for CuI, AgI,
Cu2BrI, and Ag2BrI monolayers, respectively. This is very interesting because
in 2D materials the binding energy of the first bright exciton is directly
proportional to the band gap [42, 43]. Basically, the smaller the QP band gap
is, the stronger the Coulomb screening becomes. Stronger Coulomb screening
naturally results in a smaller exciton binding energy and a larger exciton
Bohr radius. However, our results show that the substitution of the Br atoms
reduces the QP band gap of the monolayer and increases the exciton binding
energy. This is probably because the intrinsic electric field caused by
inversion symmetry breaking reduces the Coulomb screening, leading to a larger
exciton binding energy. The calculated binding energy for each monolayer
indicates the first bright exciton to be strongly confined in an area with a
Bohr radius smaller than the bond length, referring to as Frenkel-type
exciton, meaning the super stability of the excitonic states against thermal
decomposition at 300 K. The obtained optical gaps reveal the potential
application of the monolayers in the ultraviolet region.

The optical oscillator strength of the monolayers is also depicted in Fig. 5.
In spectroscopy, oscillator strength is a dimensionless quantity that
expresses the probability of absorption or emission of electromagnetic
radiation in transitions between the energy levels. Hence, direct (bright)
transitions will have large oscillator strengths. As can be seen, there are
many excitonic states with large oscillator strengths. For instance, the
oscillator strength of the first peak is 2519, 4243, 634, and 1578 for CuI,
AgI, Cu2BrI, and Ag2BrI monolayers, respectively. This means that the first
peak of the optical spectra corresponds to a bright exciton. On the contrary,
we have many dark transitions with small oscillator strength, which are known
as spin-forbidden dark excitons. Among the most important ones, one can refer
to the spin-forbidden dark exciton below the optical gap at 3.13, 2.92, 2.63,
and 2.61 eV for CuI, AgI, Cu2BrI, and Ag2BrI monolayers, respectively. These
excitons have lower energy ($\sim 0.01$ eV) than the first bright excitons.
Therefore, the ground state excitons of these monolayers are dark.

Fig. 6 provides the amplitude of transition dipole moment (TDM) of the
monolayers from the highest occupied valence band to the lowest unoccupied
conduction band. The TDM is a complex vector quantity that includes the phase
factors associated with the two states, and the amplitude of TDM gives the
probability of transition between the two states. Obviously, for all the
monolayers, the amplitude of TDM is very small in the $K\rightarrow M$ path,
implying no optical absorption between the two states in this path. Meanwhile,
the largest amplitude is located at the $\Gamma$ point, revealing the high
probability of transition between the VBM and CBM. The transition at the
$\Gamma$ point is known as allowed transition.

Figure 7: Predicted lattice thermal conductivity of the monolayers as a
function of temperature with taking the isotope scattering into account.
Figure 8: Frequency dependent phonon group velocity (left panel) and lifetime
(right panel) of CuI, AgI, and Cu2BrI monolayers at 300 K.

Owing to the dependence of the single-shot G0W0 method on the DFT mean-field
starting points, we performed the fully self-consistent GW for CuI monolayer.
As shown in Fig. S7, updating the eigenvalues and eigenstates leads to a
slight red shift in the imaginary part of the dielectric function. Also, the
intensity of the peaks is reduced, however, the location and number of the
peaks stay almost unchanged. As a result, the binding energy of the ground
state exciton remains constant at $\sim$0.87 eV. Hence, we find that the
obtained results are not dependent on the choice of input parameters.

The other optical coefficients of the monolayers i.e. the real part of the
dielectric function, absorption coefficient, and reflectivity are calculated
at the G0W${}_{0}+$BSE level. As shown in Fig. S8 (a), the static dielectric
constant of the monolayers is 1.36, 1.44, 1.29, and 1.32 for CuI, AgI, Cu2BrI,
and Ag2BrI monolayers, respectively. That is to say, with increasing the
average atomic number of the monolayer, the static dielectric constant
increases. It is also seen that the real part of the dielectric function
remains always positive in the entire range, showing the ultralow reflectivity
of the monolayers. From the absorption coefficients, it is understood that the
monolayers are good absorbers in the ultraviolet region (3.26 to 10 eV). The
mean value of absorption in this region is 1.37, 1.87, 1.16, and 1.44 $\times$
$10^{7}$ m-1 for CuI, AgI, Cu2BrI, and Ag2BrI monolayers, respectively.
Indeed, the structure with the largest (smallest) atomic number has the
largest (smallest) absorption rate. From Fig. S8 (c), it can be seen that the
monolayers are highly transparent in the visible light region (1.63 to 3.26
eV). In addition, the mean value of reflectivity in the entire region remains
under 3, 4, 2, and 2% for CuI, AgI, Cu2BrI, and Ag2BrI monolayers,
respectively. Overall, for their band gaps, ultrahigh transparency, and
flexibility, the monolayers are predicted to be very suitable for application
as electron and hole transport layers in solar cell.

###  III.3 Phononic properties

We now investigate the temperature-dependent lattice thermal conductivity of
CuI, AgI, Cu2BrI, and Ag2BrI monolayers, as it is illustrated in Fig. 7. The
results show isotropic lattice thermal conductivity in these novel monolayers,
which is consistent with their highly symmetrical lattice. At 300 K, the
thermal conductivity of CuI, AgI, Cu2BrI, and Ag2BrI monolayers taking the
isotope scattering into account are predicted to be remarkably low as 3.75,
2.27, 3.13, and 1.26 W/mK, respectively. We found that the lattice thermal
conductivity in these systems follows the $T^{-1}$ trend with respect to the
temperature ($T$). The predicted values for thermal conductivity are also
consistent with the classical theory saying a material with lower elastic
modulus and higher atomic weight shows lower thermal conductivity. By
increasing temperature up to 800 K, due to the enhanced phonon scattering, the
thermal conductivity of CuI, AgI, Cu2BrI, and Ag2BrI monolayers are reduced to
1.40, 0.85, 1.16, and 0.47 W/mK, which are very promising for thermoelectric
applications.

To better understand the underlying mechanism resulting in the low lattice
thermal conductivity of these novel 2D materials, in Fig. 8, we compare their
phonon group velocity and lifetime. As expected, with the weakening of the
atomic bonds, the phonon group velocities are considerably suppressed in the
AgI monolayer, which is more noticeable for low-frequency acoustic modes. This
is consistent with the observed narrower dispersions for the acoustic phonon
modes in AgI monolayer than those of CuI and Cu2BrI counterparts. The phonon
lifetime for these systems shows closer trends. It is noticeable that while
the phonon group velocities are close for CuI and Cu2BrI monolayers, the
phonon lifetime is generally lower, particularly between 1$-$4 THz, in Cu2BrI
monolayer than corresponding modes in the CuI counterpart, explaining its
lower thermal conductivity.

##  IV Conclusion

Motivated by the successful synthesis of novel CuI and AgI monolayers via the
graphene encapsulation approach, herein, we carried out elaborated first-
principles simulations to explore the key physical properties of non-Janus CuI
and AgI and Janus Cu2BrI and Ag2BrI monolayers. We found that these novel 2D
systems are stable, but are also soft materials with low elastic modulus. On
the basis of full-iterative solution of the Boltzmann transport equation, the
lattice thermal conductivity of CuI, AgI, Cu2BrI, and Ag2BrI monolayers at
room temperature were predicted to be remarkably low, 3.75, 2.27, 3.13, and
1.26 W/mK, respectively, which are promising for thermoelectric applications.
The electronic and optical properties were explored using many-body
perturbation calculations. Particularly, the excitonic effects on the optical
properties were taken into account by solving the Bethe-Salpeter equation. The
optical gaps were obtained to be 3.14, 2.93, 2.64, and 2.62 eV for CuI, AgI,
Cu2BrI, and Ag2BrI monolayers, which correspond to tightly bound excitons with
binding energies of 0.88, 0.92, 0.95, and 0.99 eV, respectively. Effects of
mechanical straining and electric filed on the tenability of the electronic
and optical properties were also analyzed. The presented results provide an
important and extensive vision for the key physical properties of the non-
Janus CuI and AgI and the Janus Cu2BrI and Ag2BrI monolayers and may be
valuable to the future applications of optoelectronic devices.

## Acknowledgment

B.M. and X.Z. appreciate the funding by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy within
the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). B.M. is
greatly thankful to the VEGAS cluster at the Bauhaus University of Weimar for
providing the computational resources.

## Declaration of Interests

The authors declare that they have no conflict of interest.

## References

  * Bao and Loh [2012] Q. Bao and K. P. Loh, ACS nano 6, 3677 (2012).
  * Zeng  _et al._ [2012] H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nature nanotechnology 7, 490 (2012).
  * Gutierrez  _et al._ [2013] H. R. Gutierrez, N. Perea-Lopez, A. L. Elias, A. Berkdemir, B. Wang, R. Lv, F. Lopez-Urias, V. H. Crespi, H. Terrones, and M. Terrones, Nano letters 13, 3447 (2013).
  * Fang  _et al._ [2012] H. Fang, S. Chuang, T. C. Chang, K. Takei, T. Takahashi, and A. Javey, Nano letters 12, 3788 (2012).
  * Zhang  _et al._ [2021] B. Y. Zhang, K. Xu, Q. Yao, A. Jannat, G. Ren, M. R. Field, X. Wen, C. Zhou, A. Zavabeti, and J. Z. Ou, Nature Materials 20, 1073 (2021).
  * Liu  _et al._ [2016] H.-J. Liu, J.-C. Lin, Y.-W. Fang, J.-C. Wang, B.-C. Huang, X. Gao, R. Huang, P. R. Dean, P. D. Hatton, Y.-Y. Chin, _et al._ , Advanced Materials 28, 9142 (2016).
  * Ji  _et al._ [2019] D. Ji, S. Cai, T. R. Paudel, H. Sun, C. Zhang, L. Han, Y. Wei, Y. Zang, M. Gu, Y. Zhang, _et al._ , Nature 570, 87 (2019).
  * Gogotsi and Anasori [2019] Y. Gogotsi and B. Anasori, ACS nano 13, 8491 (2019).
  * Naguib  _et al._ [2021] M. Naguib, M. W. Barsoum, and Y. Gogotsi, Advanced Materials 33, 2103393 (2021).
  * Mannix  _et al._ [2015] A. J. Mannix, X.-F. Zhou, B. Kiraly, J. D. Wood, D. Alducin, B. D. Myers, X. Liu, B. L. Fisher, U. Santiago, J. R. Guest, _et al._ , Science 350, 1513 (2015).
  * Mannix  _et al._ [2018] A. J. Mannix, Z. Zhang, N. P. Guisinger, B. I. Yakobson, and M. C. Hersam, Nature nanotechnology 13, 444 (2018).
  * Zhang  _et al._ [2016] X. Zhang, J. Hu, Y. Cheng, H. Y. Yang, Y. Yao, and S. A. Yang, Nanoscale 8, 15340 (2016).
  * Joseph  _et al._ [2020] J. Joseph, V. S. Sivasankarapillai, S. Nikazar, M. S. Shanawaz, A. Rahdar, H. Lin, and G. Z. Kyzas, ChemSusChem 13, 3754 (2020).
  * Sun  _et al._ [2012] Y. Sun, Z. Sun, S. Gao, H. Cheng, Q. Liu, J. Piao, T. Yao, C. Wu, S. Hu, S. Wei, _et al._ , Nature communications 3, 1 (2012).
  * Acharya  _et al._ [2013] S. Acharya, B. Das, U. Thupakula, K. Ariga, D. Sarma, J. Israelachvili, and Y. Golan, Nano letters 13, 409 (2013).
  * Miyake  _et al._ [1952] S. Miyake, S. Hoshino, and T. Takenaka, Journal of the Physical Society of Japan 7, 19 (1952).
  * Grundmann  _et al._ [2013] M. Grundmann, F.-L. Schein, M. Lorenz, T. Böntgen, J. Lenzner, and H. von Wenckstern, physica status solidi (a) 210, 1671 (2013).
  * Yao  _et al._ [2018] K. Yao, P. Chen, Z. Zhang, J. Li, R. Ai, H. Ma, B. Zhao, G. Sun, R. Wu, X. Tang, _et al._ , npj 2D Materials and Applications 2, 1 (2018).
  * Mustonen  _et al._ [2022] K. Mustonen, C. Hofer, P. Kotrusz, A. Markevich, M. Hulman, C. Mangler, T. Susi, T. J. Pennycook, K. Hricovini, C. Richter, _et al._ , Advanced Materials 34, 2106922 (2022).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Physical review B 54, 11169 (1996).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical review letters 77, 3865 (1996).
  * Csonka  _et al._ [2009] G. I. Csonka, J. P. Perdew, A. Ruzsinszky, P. H. Philipsen, S. Lebègue, J. Paier, O. A. Vydrov, and J. G. Ángyán, Physical Review B 79, 155107 (2009).
  * Monkhorst and Pack [1976] H. J. Monkhorst and J. D. Pack, Physical review B 13, 5188 (1976).
  * Krukau  _et al._ [2006] A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, The Journal of chemical physics 125, 224106 (2006).
  * Henkelman  _et al._ [2006] G. Henkelman, A. Arnaldsson, and H. Jónsson, Computational Materials Science 36, 354 (2006).
  * Shishkin and Kresse [2007] M. Shishkin and G. Kresse, Physical Review B 75, 235102 (2007).
  * Mostofi  _et al._ [2014] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Computer Physics Communications 185, 2309 (2014).
  * Rohlfing and Louie [2000] M. Rohlfing and S. G. Louie, Physical Review B 62, 4927 (2000).
  * Onida  _et al._ [2002] G. Onida, L. Reining, and A. Rubio, Reviews of modern physics 74, 601 (2002).
  * Mohebpour  _et al._ [2022] M. A. Mohebpour, S. M. Mozvashi, S. I. Vishkayi, and M. B. Tagani, Physical Review Materials 6, 014012 (2022).
  * Shapeev [2016] A. V. Shapeev, Multiscale Modeling & Simulation 14, 1153 (2016).
  * Novikov  _et al._ [2020] I. S. Novikov, K. Gubaev, E. V. Podryabinkin, and A. V. Shapeev, Machine Learning: Science and Technology 2, 025002 (2020).
  * Togo and Tanaka [2015] A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).
  * Mortazavi  _et al._ [2020] B. Mortazavi, I. S. Novikov, E. V. Podryabinkin, S. Roche, T. Rabczuk, A. V. Shapeev, and X. Zhuang, Applied Materials Today 20, 100685 (2020).
  * Mortazavi  _et al._ [2021] B. Mortazavi, E. V. Podryabinkin, I. S. Novikov, T. Rabczuk, X. Zhuang, and A. V. Shapeev, Computer Physics Communications 258, 107583 (2021).
  * Li  _et al._ [2014] W. Li, J. Carrete, N. A. Katcho, and N. Mingo, Computer Physics Communications 185, 1747 (2014).
  * Huang  _et al._ [2020] X. Huang, L. Yan, Y. Zhou, Y. Wang, H.-Z. Song, and L. Zhou, The Journal of Physical Chemistry Letters 12, 525 (2020).
  * Mera Acosta  _et al._ [2019] C. Mera Acosta, A. Fazzio, and G. M. Dalpian, npj Quantum Materials 4, 1 (2019).
  * He  _et al._ [2021] Z. He, R. Peng, X. Feng, X. Xu, Y. Dai, B. Huang, and Y. Ma, Physical Review B 104, 075105 (2021).
  * Min  _et al._ [2019] J. Min, C. Zhao, Z. Zeng, Y. Jia, and Z. Du, Physical Review B 100, 085402 (2019).
  * Xu  _et al._ [2017] L. Xu, M. Yang, S. J. Wang, and Y. P. Feng, Physical Review B 95, 235434 (2017).
  * Long  _et al._ [2021] C. Long, Y. Dai, and H. Jin, Physical Review B 104, 125306 (2021).
  * Jiang  _et al._ [2017] Z. Jiang, Z. Liu, Y. Li, and W. Duan, Physical review letters 118, 266401 (2017).