# Magnetoexcitons in transition-metal dichalcogenides monolayers, bilayers,
and van der Waals heterostructures

Roman Ya. Kezerashvili1,2 and Anastasia Spiridonova1,2 E-mail contact: 1The
Graduate School and University Center, The City University of New York, New
York, NY 10016, USA
2New York City College of Technology, The City University of New York,
Brooklyn, NY 11201, USA

###### Abstract

We study direct and indirect magnetoexcitons in Rydberg states in monolayers
and heterostructures of transition-metal dichalcogenices (TMDCs) in an
external magnetic field, applied perpendicular to the monolayer or
heterostructures. We calculate binding energies of magnetoexcitons for the
Rydberg states 1$s$, 2$s$, 3$s$, and 4$s$ by numerical integration of the
Schrödinger equation using the Rytova-Keldysh potential for direct
magnetoexcitons and both the Rytova-Keldysh and Coulomb potentials for
indirect magnetoexcitons. Latter allows understanding the role of screening in
TMDCs heterostructures. We report the magnetic field energy contribution to
the binding energies and diamagnetic coefficients (DMCs) for direct and
indirect magnetoexcitons. The tunability of the energy contribution of direct
and indirect magnetoexcitons by the magnetic field is demonstrated. It is
shown that binding energies and DMCs of indirect magnetoexcitons can be
manipulated by the number of hBN layers. Therefore, our study raises the
possibility of controlling the binding energies of direct and indirect
magnetostrictions in TMDC monolayers, bilayers, and heterostructures using
magnetic field and opens an additional degree of freedom to tailor the binding
energies and DMCs for heterostructures by varying the number of hBN sheets
between TMDC layers. The calculations of the binding energies and DMCs of
indirect magnetoexcitons in TMDC heterostructures are novel and can be
compared with the experimental results when they will be available.

##  I Introduction

Transition-metal dichalcogenices (TMDCs) present a great interest for
application in nano- and quantum devices at room temperature owing to Dirac
cone [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], direct gap at nonequivalent points
$K/K^{\prime}$ [11, 12, 13], and strong-spin orbit interaction that lifts
degeneracy at valence and conduction bands [14, 15, 16]. In design of
electronic devices, TMDCs monolayers, TMDC/TMDC, and TMDC/insulator/TMDC
heterostructures can be used. Moreover, TMDCs present a great interest for
fundamental research. TMDCs heterostructure can support high-temperature
quantum Bose gases of indirect excitons [17, 18, 19, 20], superfluidity of
electron-hole [21, 22] and dipolar excitons [23], and superconductivity [24].

TMDCs are semiconductors that have a chemical formula MX2 where M is a
transition metal atom (Mo or W) and X is a chalcogen atom (S, Se, or Te). The
comprehensive review of excitonic complexes in TMDCs monolayers is given in
Refs. [25, 14, 26]. Excitons in monolayers have a direct energy gap. However,
TMDCs heterostructures made of monolayers have indirect gaps. There are two
possible arrangements of double-layer heterostructures: bilayer TMDC-TMDC [27,
28, 29, 30, 31, 32, 33, 34, 19, 35, 36, 37, 20] and TMDC-insulator-TMDC [17,
38, 35]. For both arrangements, there are two types of heterostructures.
Type-I heterostructure is a heterostructure formed by the same type of TMDC
monolayers [39, 29, 19, 35], and type-II heterostructure is formed by
monolayers of different TMDC monolayers [27, 28, 33, 30, 40, 23, 32, 31, 34,
36, 35, 37, 20]. Both type-I and II heterostructures have indirect band gap
[41, 42, 39, 43]. However, stacking order [44, 39] or application of strain
[43] can be used to induce indirect gap into direct gap. In addition, the
application of perpendicular electric field controlled by voltage modifies the
band structure so it is advantageous for electrons and holes to reside in
different TMDC layers [17, 19, 45] and makes indirect excitons more preferable
than direct excitons. However, the application of the electric field can lead
to dissociation of indirect excitons [46]. Nevertheless, the extended lifetime
of indirect excitons [28, 29, 47, 34] compared to direct excitons presents
great interest for the design of electronic devices.

The history of studying excitons in semiconductors in the magnetic field
extends back for 60 years. Starting from Elliot and Loudon [48] and Hasegawa
and Howard [49], who developed the theory of the Mott exciton in the strong
magnetic field followed by authors of Refs. [50, 51, 52], who studied the
physics of Mott magnetoexcitons. For direct magnetoexcitons in TMDCs, the
binding energies of Rydberg states of direct excitons are reported in Refs.
[53, 54, 55, 56, 57, 58, 59], the Zeeman (ZM) shift has been considered in
Refs. [60, 61, 62, 63, 54, 64, 65, 66, 56, 67, 68, 58, 57, 69], and the
diamagnetic (DM) shift was addressed in Refs. [70, 60, 63, 71, 54, 64, 56, 66,
72, 55, 73, 68, 57, 58, 74]. For indirect excitons in double-layer
heterostructures Rydberg states binding energies are reported in [39, 28, 29,
30, 40, 75, 76, 35, 34, 77, 46, 20], and Zeeman shift is addressed in Refs.
[33, 78, 79, 80, 81, 82].
Historically, the electromagnetic interaction between electron and hole was
described by Coulomb potential ($V_{C}$) for both direct and indirect excitons
[48, 50, 51, 52, 83, 84, 85, 86, 87, 88]. However, as has been shown, the
Rytova-Keldysh (RK) potential ($V_{RK}$) [89, 90] is the appropriate potential
to describe the interaction of electron and hole in two-dimensional (2D)
configuration space since the RK potential takes into account the effects of
the dielectric environment and the 2D confinement. As a result, currently,
both the RK [26, 57, 54, 68, 65, 58, 72, 55, 59, 91, 92, 93, 94] and Coulomb
potential are used to describe the direct exciton. For indirect excitons, the
Coulomb potential is used. However, in a few cases, the RK potential is used
as well. We do the same thing: we use both the RK and Coulomb potentials for
indirect exciton to investigate the importance of using the accurate potential
that describes an interaction between hole and electron.
In this paper, we study the effects of the external magnetic field on the
binding energies of 1$s$, 2$s$, 3$s$, and 4$s$ Rydberg states of direct and
indirect magnetoexcitons in TMDCs. $A$ and $B$ direct magnetoexcitons are
considered in freestanding (FS) and encapsulated by the hexagonal boron
nitride (hBN) MX2 monolayers, and $A$ and $B$ indirect magnetoexcitons are
considered in bilayer MX2-MX2 and MX2-hBN-MX2 type-I heterosturctures. In our
study to find eigenfunctions and eigenenergies for indirect magnetoexcitons, a
two-particle Schrödinger equation is solved using the Rytova-Keldysh [89, 90]
and Coulomb potentials to calculate binding energies and diamagnetic
coefficients. We demonstrate strong sensitivity of the energy contributions
from the external magnetic field on the type of the potential. The first time
diamagnetic coefficients (DMCs) for bilayer MX2-MX2 and MX2-hBN-MX2
heterostructures are calculated and reported. Our study raises the possibility
to control the binding energies of direct and indirect magnitoexcitons in
monolayer MX2, bilayer MX2-MX2, and MX2-hBN-MX2 heterostructures using a
magnetic field and opens an additional degree of freedom to tailor the binding
energies and DMCs for heterostructures by varying the number of hBN sheets
between TMDC layers.

The article is structured as follows. In Sec. II we introduce the theoretical
formalism for the description of Mott-Wannier magnetoexcitons in freestanding
and encapsulated MX2 monolayers, and in TMDC heterostructures and discuss
electrostatic interactions that form direct and indirect magnetoexcitons. In
Sec. III we focus on the analysis and discussion of the results. In
particular, contributions from the external magnetic field to binding energies
of magnetoexcitons in MX2 monolayers, MX2-MX2 bilayers, and MX2-hBN-MX2
heterostructures are presented and discussed in Subsecs. III.1 and III.2,
respectively. Subsec. III.3 is devoted to the calculations of diamagnetic
coefficients for direct and indirect magnetoexcitons. Finally, Sec. IV
concludes upon the results.

##  II Theoretical Formalism

In this section, we provide the theoretical formalism for describing the Mott-
Wannier magnetoexciton in 2D materials and present the energy contribution
from the external magnetic field to the Rydberg states binding energy of
magnetoexcitons. Here, we present electrostatic interactions that due to a
non-local dielectric screening of an electron-hole interaction strongly modify
the electrostatic Coulomb potential and form direct magnetoexcitons in TMDC
monolayers and indirect magnitoexcitons in MX2-MX2 bilayers and MX2-hBN-MX2
van der Waals heterostructure, where $N$ layers of hBN monolayers separate two
TMDC monolayer

###  II.1 Mott-Wannier magnetoexciton in 2D materials

Let us introduce the equation for the description of Mott-Wannier excitons in
the external magnetic field. We employ an effective mass model for two charged
point-like particles in two dimensions. In general, electrostatically-bound
electrons and holes in the external magnetic field form magnetoexcitons. To
find the eigenfunctions and eigenenergies of a 2D magnetoexciton in TMDCs
monolayer, bilayer MX2-MX2, and MX2-hBN-MX2 heterostructure in the external
magnetic field, we write the Schrödinger equation for an interacting electron
and hole [85] $(\hbar=c=1)$:

$\bigg{[}\frac{1}{2m_{e}}\Big{(}-i\nabla_{e}+e\mathbf{A(r}_{e})\Big{)}^{2}+\frac{1}{2m_{h}}\Big{(}-i\nabla_{h}-e\mathbf{A(r}_{h})\Big{)}^{2}+V\left(r_{e},r_{h}\right)\bigg{]}\psi\left(\mathbf{r}_{e},\mathbf{r}_{h}\right)=E\psi\left(\mathbf{r}_{e},\mathbf{r}_{h}\right),$
(1)

where $e$ is the charge of the electron, the indices $e$ and $h$ are referring
to the electron and hole, respectively, $\mathbf{r}_{e}$ and $\mathbf{r}_{h}$
are 2D coordinates and $m_{e}$ and $m_{h}$ are the masses of charged
particles, $\mathbf{A(r}_{e/h})=\mathbf{B\times r}_{e/h}/2$ is a gauge vector
potential, and $V\left(r_{e},r_{h}\right)$ is the potential of interaction
between the electron and hole confined in 2D space. In three-dimensional (3D)
homogeneous dielectric environments, the electron-hole interaction is
described by the Coulomb potential. However, in 2D monolayer this interaction
has to be modified, because of the reduced dimensionality and screening
effects. The original derivation of two charged particles interaction in 2D
space was given by Rytova [89] and a decade later was independently obtained
by Keldysh [90]. The potential is called the Rytova-Keldysh (RK) potential.
For almost a decade, the RK potential has been used to describe electrostatic
interaction between charge carriers of few-body complexes in transition-metal
dehydrogenizes, phosphorene, and Xenes monolayers [26] and references herein.

Following Refs. [48, 51, 50, 52, 85, 88], in Eq. (1) we introduce the
coordinate of the center-of-mass
$\mathbf{R}=\frac{m_{e}\mathbf{r}_{e}-m_{h}\mathbf{r}_{h}}{M}$ and the
relative motion coordinate $\mathbf{r}=\mathbf{r}_{e}-\mathbf{r}_{h},$ where
the total mass of the system is $M=m_{e}+m_{h}$, and consider the magnetic
field pointing in $z$-direction that is perpendicular to the monolayer where
the exciton is located, $\mathbf{B}=\mathbf{B}(0,0,B)$. After performing the
standard procedure for the coordinate transformation to the center-of-mass,
Eq. (1) becomes:

$\displaystyle\bigg{[}$ $\displaystyle-$
$\displaystyle\frac{1}{2M}\frac{{\partial}^{2}}{\partial\bm{R}^{2}}-\frac{1}{2\mu}\frac{{\partial}^{2}}{\partial\bm{r}^{2}}+\frac{e^{2}}{8\mu}(\bm{B}\times\bm{R})^{2}+\frac{e^{2}\mu^{2}}{8}\left(\frac{1}{m_{e}^{3}}+\frac{1}{m_{h}^{3}}\right)(\bm{B}\times\bm{r})^{2}-\frac{ie}{2M}(\bm{B}\times\bm{r})\cdot\frac{\partial}{\partial\bm{R}}-$
(2)
$\displaystyle-\frac{ie}{2\mu}(\bm{B}\times\bm{R})\cdot\frac{\partial}{\partial\bm{r}}-\frac{ie\gamma}{2\mu}(\bm{B}\times\bm{r})\cdot\frac{\partial}{\partial\bm{r}}+\frac{e^{2}\gamma}{4\mu}(\bm{B}\times\bm{R})\cdot(\bm{B}\times\bm{r})+V(\bm{r})\bigg{]}\psi(\bm{R},\bm{r})=E\psi(\bm{R},\bm{r}),$

where $\gamma=\frac{m_{h}-m_{e}}{m_{h}+m_{e}}$ and
$\mu=\frac{m_{e}m_{h}}{m_{e}+m_{h}}$ is the reduced mass. Equation (2) is
written for the case when masses of the electron and hole are different:
$m_{h}\neq m_{e}$, which is the case in TMDCs. The term
$(\bm{B}\times\bm{r})\cdot\frac{\partial}{\partial\bm{r}}=\bm{B}\cdot\bm{L}=0$
since we consider Rydberg states: 1$s$, 2$s$, 3$s$, and 4$s$, for which $l=0$,
$m_{l}=0$.

Following Refs. [51, 88, 85], we introduce an operator $\hat{\bm{P}}$:

$\hat{\bm{P}}=-i\nabla_{\bm{R}}-\frac{e}{2}(\bm{B}\times\bm{r}).$ (3)

It is easy to check that $\hat{\bm{P}}$ commutes with the Hamiltonian in Eq.
(1), therefore, it has the same eigenfunction as Eq. (1). Thus, one can write
the wave function for the exciton in the magnetic field as [51, 88]:

$\psi(\bm{R},\bm{r})=e^{\left[i\bm{R}\cdot(\bm{P}+\frac{e}{2}\bm{B}\times\bm{r})\right]}e^{\frac{1}{2}i\gamma\bm{r}\cdot\bm{P}}\Phi(\bm{r}-\tilde{\bm{\rho}}_{0}),$
(4)

where the notation
$\tilde{\bm{\rho}}_{0}=\frac{1}{eB^{2}}(\bm{B}\times\bm{P})$ is introduced.
After substituting the wave function $\psi(\bm{R},\bm{r})$ in Eq. (2), the
Schrödinger equation for the relative motion of the electron and hole reads:

$\left[\frac{\bm{P}^{2}}{4\mu}+\frac{e}{4\mu}\bm{P}\cdot(\bm{B\times\bm{r}})-\frac{1}{2\mu}\frac{{\partial}^{2}}{\partial\bm{r}^{2}}+\frac{e^{2}}{8\mu}(\bm{B}\times\bm{r})^{2}-\frac{ie\gamma}{2\mu}(\bm{B}\times\bm{r})\cdot\frac{\partial}{\partial\bm{r}}-\frac{i\gamma}{2\mu}\bm{P}\frac{\partial}{\partial\bm{r}}+V(r)\right]\Phi=E\Phi.$
(5)

Finally, the equation for a 2D electron-hole pair with zero center-of-mass
momentum reads [86, 88]:

$\left[-\frac{1}{2\mu}\frac{{\partial}^{2}}{\partial\bm{r}^{2}}+\frac{e^{2}}{8\mu}(\bm{B}\times\bm{r})^{2}+V(r)\right]\Phi(r)=E\Phi(r).$
(6)

Equation (6) describes the Mott–Wanner magnetoexciton in Rydberg optical
states in 2D materials. This equation has a long history of the solution in
the case of the electron-hole Coulomb interaction [48, 51, 50, 52, 83, 84, 85,
86, 87, 88]. However, we solve Eq. (6) using $V_{RK}$ for direct
magnetoexcitons. For indirect magnetoexcitons, both $V_{RK}$ and $V_{C}$
potentials are used to investigate the importance of using correct potential
describing electron-hole interactions. Note that Eq. (6) does not explicitly
contain any spin- or valley-dependent Zeeman terms. Therefore, the
eigenenergies correspond to the average transition energy of each exciton
state between points $K$ and $K^{\prime}$ [56].

By calculating the magnetoexciton energy of Rydberg states at different values
of the magnetic field, we can find the energy contribution from the magnetic
field to the binding energy of a magnetoexciton in the following way:

$\Delta E=E_{0}-E(B).$ (7)

In Eq. (7) $E_{0}$ is the exciton energy when the magnetic field is absent,
while $E(B)$ is the magnetoexciton energy at some value of the magnetic field.

In literature, the Zeeman and diamagnetic shifts of excitons in TMDCs
monolayers are treated in the same way as ZM and DM shifts of excitons in
quantum wells and quantum dots [95, 96, 97, 98, 99, 100, 101, 102, 103]. The
first step is to write a magnetoexciton energy using Taylor series [97]:

$E(B)=E_{0}+\gamma_{1}B+\gamma_{2}B^{2}+...$ (8)

In Eq. (8) the second and third terms are the ZM and DM shifts, respectively.
When the magnetic field’s energy contribution is small compared to the binding
energy in the absence of the magnetic field, we can use the first three terms
of Taylor series to describe the magnetoexciton binding energy [97, 99, 101,
68]. In other words, the following condition $E_{0}>|E(B)-E_{0}|$ is met.
However, when $E_{0}\sim|E(B)-E_{0}|$, higher-order terms in Eq. (8) need to
be considered.

As stated before, the treatment of the ZM and DM shifts of magnetoexcitons in
TMDCs monolayers follows the same procedure applied for quantum dots and
quantum wells. The second step in describing ZM and DM shifts is to define
them by connecting shifts to terms of Taylor series. The Zeeman shift is
defined as $\gamma_{1}B=-\mu_{B}gB$ [102, 99, 101, 60, 63, 65, 67, 68], where
$g$ is the $g$ factor and $\mu_{B}$ is Bohr magneton. Experimentally, the ZM
shift is defined as energy difference between $K$/$K^{\prime}$ points i.e.
$-g\mu_{B}B=E(K)-E(K^{\prime})$ [100, 101, 54, 64, 65, 56, 67, 58, 57, 68].
The diamagnetic shift is defined as $\gamma_{2}B^{2}=\frac{e^{2}}{8\mu}\langle
r^{2}\rangle B^{2}$ [95, 96, 97, 102, 98, 99, 100, 101, 103, 54, 56, 58, 57,
68], where $\langle r^{2}\rangle$ is the expectation value of $r^{2}$ over the
exciton envelope wave function. In our work, we follow notation used for
magnetoexciton diamagnetic coefficient in TMDCs monolayers i.e.
$\gamma_{2}=\sigma$ [63, 54, 56, 67, 57, 58]. According to Refs. [95, 96, 97,
102, 98, 99, 100, 101, 103, 54, 56, 67, 58, 57] the DMC for a carrier in a
semiconductor is defined as $\sigma=\frac{e^{2}}{8\mu}\langle r^{2}\rangle$.
Therefore, by calculating DMCs, exciton radius, reduced mass, and dielectric
properties of a material can be obtained [97, 54, 56, 57, 58]. Experimentally,
the DM shift is defined as $\sigma B^{2}=\frac{E(K)+E(K^{\prime})}{2}$ [99,
103, 101, 57, 60, 68].

###  II.2 Magnetoexcitons in TMDCs monolayer

A non-local dielectric screening of an electron-hole interaction strongly
modifies the electrostatic Coulomb potential. It leads to a non-hydrogenic
Rydberg series of exciting magnetoexciton states. The Rytova-Keldysh potential
has become a prevalent description of the electrostatic interaction of charge
carriers in 2D systems. The RK potential is central potential, and the
interaction between the electron and hole for direct excitons in an
encapsulated TMDC monolayer has the form [89, 90]:

$V_{RK}(r)=-\frac{\pi
ke^{2}}{2\kappa\rho_{0}}\left[H_{0}\left(\frac{r}{\rho_{0}}\right)-Y_{0}\left(\frac{r}{\rho_{0}}\right)\right],$
(9)

where $r\equiv r_{eh}=r_{e}-r_{h}$ is the relative coordinate between the
electron and hole. In Eq. (9), $\kappa=(\epsilon_{1}+\epsilon_{2})/2$
describes the surrounding dielectric environment, $\epsilon_{1}$ and
$\epsilon_{2}$ are the dielectric constants below and above the monolayer,
$H_{0}$ and $Y_{0}$ are the Struve and Bessel functions of the second kind,
respectively, and $\rho_{0}=2\pi\chi_{2D}/\kappa$ [104] is the screening
length, where $\chi_{2D}$ is the 2D polarizability of the monolayer, which is
given by $\chi_{2D}=h\epsilon/4\pi$ [90], where $\epsilon$ is the bulk
dielectric constant of the exciton/hole containing monolayer and $h$ is the
thickness of a TMDC monolayer. At $r<\rho_{0}$ the RK potential has
logarithmical dependence and diverges logarithmically at the origin. At long-
range distances, $r>\rho_{0}$, it retains the Coulomb potential behavior.
Using the RK potential for description of excitons in 2D monolayer, one finds
a strong dependence of the binding energy on whether the monolayer is
suspended in air (Fig. 1$a$), encapsulated by hBN (Fig. 1$b$), or in
heterostructures (Figs. 1$c$ and 1$d$). It is worth mentioning that the RK
potential is obtained for a monolayer. However, a TMDC monolayer consists of
three atomic layers: chalcogen-metal-chalcogen sheets. To address this
problem, a new potential is derived in Ref. [105] which considers the three
atomic sheets that compose a monolayer of TMDC and can explain the
nonhydrogenic Rydberg series of excitons in TMDC monolayer. The Rytova-Keldysh
potential can be recovered when considering the strict 2D limit, and in our
study, we are using the RK potential.

Figure 1: (Color online) Schematic illustration of magnetoexcitons in TMDC
monolayers and heterostructures. $(a)$ A direct magnetoexciton in a
freestanding TMDC monolayer. $(b)$ A direct magnetoexciton in an encapsulated
TMDC monolayer. $(c)$ An indirect magnetoexciton in a freestanding bilayer
MX2-MX2 heterostructure. $(d)$ An indirect magnetoexciton in MX2-hBN-MX2 van
der Waals heterostructure.

###  II.3 Indirect magnetoexcitons in TMDCs heterostructures

Van der Waals-like forces can hold together layered TMDC materials. When two
TMDC monolayers are brought together, the conduction band minimum, and the
valence band maximum reside in two different layers. Electrons and holes in
such a structure prefer to reside in separate layers [46]. Let us now consider
the formation of indirect magnetoexcitons formed by electrons and holes in
type-I heterostructure molded by the same type of TMDC monolayers: a bilayer
MX2-MX2 heterostructure, shown in Fig. 1$c$ and van der Waals MX2-hBN-MX2
heterostructure, where two TMDC monolayers are separated by $N$ layers of hBN
monolayers, presented in Fig. 1$d$. Such magnetoexcitons have a longer
lifetime than the direct excitons due to longer recombination time. In such
structures, the electron and hole move in the planes of different layers, form
an indirect exciton, and have restricted motion between the layers due to the
dielectric barrier. The latter is a two-body problem in the restricted 3D
space, and the electrostatic electron-hole interaction $V(r)$ could form a
bound state, i.e., the indirect magnetoexciton. Therefore, to determine the
binding energy of the magnetoexciton one must solve a two-body problem in
restricted 3D space because the motion in $z$-direction is restricted. The
relative separation $r$ between the electron and hole can be written in
cylindrical coordinates as $\bm{r}=\rho\hat{\bm{\rho}}+D\hat{\bm{z}}$, where
$\hat{\bm{\rho}}$ and $\hat{\bm{z}}$ are unit vectors. Writing $r$ in
cylindrical coordinates allows us to treat the case of direct excitons in a
TMDC monolayer and spatially indirect excitons in bilayer MX2-MX2 and MX2-hBN-
MX2 heterostructures on equal footing. If we set $D=0$ in the latter
expression, it becomes a purely 2D equation, with $\rho$ representing the
separation between the electron and hole sharing the same plane. For spatially
indirect excitons, the relative distance between the electron and hole is
$r=\sqrt{\rho^{2}+D^{2}}$, where $D$ is the distance between the middle of two
TMDC layers assuming that the electron and hole reside in the middle of their
respective sheets [46].

Thus, for the description of indirect Mott–Wanner magnetoexcitons one can use
Eq. (6) with interactions

$V_{RK}(\sqrt{\rho^{2}+D^{2}})=-\frac{\pi
ke^{2}}{2\kappa\rho_{0}}\left[H_{0}\left(\frac{\sqrt{\rho^{2}+D^{2}}}{\rho_{0}}\right)-Y_{0}\left(\frac{\sqrt{\rho^{2}+D^{2}}}{\rho_{0}}\right)\right]$
(10)

for the RK potential, and

$V_{C}\left(\sqrt{\rho^{2}+D^{2}}\right)=-\frac{ke^{2}}{\kappa\left(\sqrt{\rho^{2}+D^{2}}\right)}$
(11)

for the Coulomb potential. Equations (10) and (11) describe the interaction
between the electron located in one and hole in the other parallel TMDC
monolayers. Therefore, one can obtain the eigenfunctions and eigenenergies of
magnetoexcitons by solving Eq. (6) using the potential (9) for direct
magnetoexcitons, or using either potential (10) or (11) for indirect
magnetoexcitons. For indirect magnetoexcitons, we perform calculations using
both the RK and Coulomb potentials. This allows a better understanding of the
importance of the screening effect in MX2-MX2 and MX2-hBN-MX2
heterostructures.

It is worth mentioning that the RK potential was originally formulated as an
explicitly 2D description of the Coulomb interaction. Nevertheless, there have
been recent attempts to apply the RK potential to indirect excitons in van der
Waals heterostructures of 2D materials such as the TMDCs, phosphorene, and
Xenes [17, 18, 106, 23, 77, 107, 108]. The logic behind considering the RK
potential for indirect excitons follows from two considerations: i. the
dielectric environment is still inhomogeneous, just as in the case of the
direct exciton – when the interlayer separation $D$ is smaller than, or
comparable to, the RK potential screening length $\rho_{0}$ and the excitonic
gyration radius $\sqrt{\langle r^{2}\rangle}$, the electron-hole interaction
potential must account for both the TMDC monolayers and the interlayer
dielectric, and ii. as the interlayer separation $D$ becomes larger than
$\rho_{0}$, the total separation between the electron and hole,
$r=\sqrt{\rho^{2}+D^{2}}$, becomes much larger than $\rho_{0}$, and,
therefore, the RK potential converges towards the Coulomb potential. Let us
emphasize that we are not claiming definitively that the RK potential provides
the most accurate description of the spatially indirect exciton, hence, we
present the extensive comparison with the corresponding results calculated
using the Coulomb potential.

Above, we consider ideal cases when encapsulated TMDC monolayer, bilayer
MX2-MX2, and MX2-hBN-MX2 heterostructures are fabricated so that each layered
material is stacked without any vacuum or air interlayer gap. In reality,
between the layers always exists non-vanishing interlayer gap. In each layer
the field lines of the Coulomb interaction are screened by the adjacent
material, which reduces the single-particle band gap as well as exciton
binding energies [109]. In Ref. [109] the authors demonstrate and give a
quantitative understanding that the binding energy of excitons and electronic
and optical properties are sensitive to the interlayer distances on the atomic
scale.

Throughout this paper, we consider the separation between the electron and
hole residing in two TMDC monolayers of the MX2-hBN-MX2 van der Waals
heterostructures in steps of calibrated thickness,
$l_{\text{hBN}}=0.333~{}\text{nm}$, corresponding to the thickness of one hBN
monolayer: $D=h+Nl_{\text{hBN}}$, where $h$ is the TMDC monolayer thickness
and $N$ is the number of hBN monolayers. For a bilayer MX2-MX2 heterostructure
$N=0$ and following Ref. [46] the electron and hole reside in the middle of
their respective sheets and $r=\sqrt{\rho^{2}+h^{2}}$.

##  III Results of calculations and discussion

We report the energy contribution from the external magnetic field to the
binding energies of $A$ and $B$ magnetoexcitons in Rydberg optical states
1$s$, 2$s$, 3$s$, and 4$s$, in freestanding (Figs. 1$a$) and encapsulated
(Figs. 1$b$) WSe2, WS2, MoSe2, and MoS2 monolayers, bilayer MX2-MX2 (Fig.
1$c$), and MX2-hBN-MX2 heterostructures (Fig. 1$d$). The diamagnetic
coefficients for the bilayer MX2-MX2 and MX2-hBN-MX2 heterostructures are
reported the first time. We are considering the further knob to tailor the
binding energies and diamagnetic coefficients for the MX2-hBN-MX2
heterostructures by varying the number of hBN sheets between TMDC layers and
present corresponding calculations.

Table 1: Parameters for the $A$ and $B$ excitons. For the $A$ exciton for each material two values of $\mu$ and $\chi_{2D}$ are given. The average dielectric constant, $\kappa=\frac{\varepsilon_{1}+\varepsilon_{2}}{2}=4.89$, is taken from Ref. [17]. $\kappa$ is the same for all materials. $\mu$ is in units of electron mass, $m_{0}$.The 2D polarizability $\chi_{2D}$ and TMDC monolayer thickness $h$ are given in Å. |  |  $\mu$ ($m_{0}$) |  $\chi_{2D}$ (Å) |  $\rho_{0}$ (Å) |  $h$ (Å) |  |  |  $\mu$ ($m_{0}$) |  $\chi_{2D}$ (Å) |  $\rho_{0}$ (Å) |  $h$ (Å)
---|---|---|---|---|---|---|---|---|---|---|---
WSe2 |  $E_{A_{\text{high}}}$ |  0.27 [110] |  7.18 [104] |  9.226 |  6.575 [53] |  MoSe2 |  $E_{A_{\text{high}}}$ |  0.31 [110] |  8.23 [104] |  10.575 |  6.527 [53]
|  $E_{A_{\text{low}}}$ |  0.15 [111] |  7.571 [53] |  9.728 |  |  |  $E_{A_{\text{low}}}$ |  0.27 [104] |  8.461 [53] |  10.872
|  $E_{B}$ |  0.16 [53] |  7.18 [104] |  9.226 |  |  |  $E_{B}$ |  0.29 [53] |  8.23 [104] |  10.575
WS2 |  $E_{A_{\text{high}}}$ |  0\. 23 [110] |  6.03 [104] |  7.748 |  6.219 [53] |  MoS2 |  $E_{A_{\text{high}}}$ |  0.28 [110] |  6.60 [104] |  8.480 |  6.18 [53]
|  $E_{A_{\text{low}}}$ |  0.15 [111] |  6.393 [53] |  8.214 |  |  |  $E_{A_{\text{low}}}$ |  0.16 [92] |  7.112 [53] |  9.138 |
|  $E_{B}$ |  0.15 [53] |  6.03 [104] |  7.748 |  |  |  $E_{B}$ |  0.24 [53] |  6.60 [104] |  8.480 |

In our calculations, we vary the magnetic field in increment of 1 T in the
range from 0 T to 30 T and use input parameters given in Table 1. In Table 1,
the values of $E_{A_{\text{high}}}$ and $E_{A_{\text{low}}}$ correspond to the
parameters found in the literature, which minimizes and maximizes the $A$
exciton binding energy, respectively. It is highly likely that each
parameter’s true value for a given material falls somewhere within the range
given. Therefore, the true magnitude of the calculated quantities for $A$
excitons studied in this paper lies somewhere between the calculated values.
Values of $E_{B}$ correspond to parameters given in the literature for $B$
excitons.

###  III.1 Contribution from the external magnetic field to binding energies
of magnetoexcitons in a monolayer

Figure 2: Energy contribution from the magnetic field to the binding energies
of Rydberg states 1$s$ ($a$), 2$s$ ($b$), 3$s$ ($c$), and 4$s$ ($d$) for
direct $A_{\text{low}}$ magnetoexcitons in FS and encapsulated by hBN MX2
monolayers. The solid and dashed curves present results of calculations for
freestanding and encapsulated TMDC monolayers, respectively.

The results of the energy contribution from the magnetic field to the binding
energy for direct $A$ and $B$ of Rydberg states magnetoexcitons in
freestanding and encapsulated TMDC monolayers are reported in Fig. 2. The
comparative analysis of the results presented in Fig. 2 shows the following:
i. the energy contribution from the magnetic field to the binding energy of
the direct magnitoexcitons in FS monolayers of TMDC materials is always less
than in encapsulated monolayers, and the difference increases with the
increase of the magnetic field; ii. the direct magnetoexcitons in FS
monolayers are bound in 1$s$, 2$s$, 3$s$, and 4$s$ states within all
considered range of the magnetic field, while magnetoexcitons in encapsulated
monolayers dissociated in the states 3$s$ and 4$s$ when the magnetic field is
greater than some particular values. These conclusions are related to the
direct $A_{\text{low}}$ magnetoexcitons. Interestingly enough that these
conclusions stay the same for direct $A_{\text{high}}$ and $B$
magnetoexcitons. However, the $\Delta E$ is systematically smaller for the
$A_{\text{high}}$, while $\Delta E$ for $B$ magnetoexcitons are lying between
$\Delta E$ for the $A_{\text{low}}$ and $A_{\text{high}}$ magnetoexcitons.
Analysis of the results shows that the energy contribution from the external
magnetic field to the magnetoexciton binding energy depends on material
parameters. In fact, the binding energy of magnetoexcitons is a function of
the reduced mass $\mu$ and polarizability $\chi_{2D}$: the binding energy of a
magnetoexciton is bigger for the smaller $\mu$ and larger $\chi_{2D}$. This
pattern qualitatively coincides with previously reported calculations for
magnetoexcitons in WSe2 and MoSe2 monolayers [74]. It is worth noting that for
MoS2 the energy contribution from the magnetic field for the state 1$s$ is
more than one order of magnitude smaller than for the states 2$s$, 3$s$ and
4$s$. Another distinct feature is that magnetoexcitons are ionized in states
3$s$ and 4$s$ at some of the magnetic field values while staying bound in
states 1$s$ and 2$s$ at higher magnetic field values.

###  III.2 Contribution from the external magnetic field to binding energies
of magnetoexcitons in bilayer MX2-MX2 and MX2-hBN-MX2 heterostructures

Figure 3: Energy contribution from the magnetic field to the binding energies
of Rydberg states 1$s$ ($a$), 2$s$ ($b$), 3$s$ ($c$), and 4$s$ ($d$) for
indirect magnetoexcitons in FS MoS2-MoS2 bilayer. The binding energy is
calculated with $V_{RK}$ (dashed lines) and $V_{C}$ (solid lines) potentials.
Data are plotted for $A_{\text{high}}$, $A_{\text{low}}$, and $B$
magnetoexcitons.

As the first step, we compare the contributions to binding energies of
indirect magnetoexcitons due to the external magnetic field calculated using
the RK and Coulomb potentials. In Fig. 3 are shown the energy contributions
from the magnetic field to the binding energies of indirect magnetoexcitons in
FS bilayer MoS2 -MoS2 for Rydberg states 1$s$ (3$a$), 2$s$ (3$b$), 3$s$
(3$c$), and 4$s$ (3$d$). The binding energies are calculated using $V_{RK}$
and $V_{C}$ potentials. Data are plotted for $A_{\text{high}}$,
$A_{\text{low}}$, and $B$ magnetoexcitons. The comparison shows that i.
$\Delta E_{RK}>\Delta E_{C}$ and the difference increases with the magnetic
field increase; ii. results obtained with both potentials satisfy the
following inequality: $\Delta E_{A_{\text{high}}}<\Delta E_{B}<\Delta
E_{A_{\text{low}}}$. Most importantly, it is worth to mention that the
contribution from the magnetic field is systematically bigger for the FS
bilayer MoS2-MoS2 than for the FS MoS2 monolayer for all states when
calculations are performed with $V_{RK}$. In other words, $\Delta
E_{A_{\text{low}}}$ for indirect magnetoexcitons in FS MoS2-MoS2 bilayer is
greater than $\Delta E_{A_{\text{low}}}$ for direct magnetoexcitons in FS MoS2
monolayer. Similar patterns are observed for other FS bilayers constituent
from MoSe2, WS2 and WSe2 monolayers, respectively.

Now let us consider indirect magnetoexcitons in MoS2-hBN-MoS2 heterostructure.
In Fig. 4 are presented the results of calculations for the dependence of the
ratio $\Delta E_{RK}/\Delta E_{C}$ on the magnetic field and number of the hBN
layers separating two parallel TMDC layers. Calculations are performed for $A$
magnetoexcitons. The ratio weakly depends on the magnetic field and converges
towards 1 when the number of hBN layers increases. The RK potential always
gives the larger contribution than the Coulomb potential. However, when the
number of hBN layers is more than three, the difference becomes very small.
Such a result is understandable because the potential
$V_{RK}(\sqrt{\rho^{2}+D^{2}})$ converges to
$V_{C}\left(\sqrt{\rho^{2}+D^{2}}\right)$ when $D$ increases.

Figure 4: Ratio of $\Delta E_{RK}/\Delta E_{C}$ for $A_{\text{high}}$
magnetoexcitons in MoS2-hBN-MoS2 heterostructure for Rydberg states 1$s$
$(a)$, 2$s$ $(b)$, 3$s$ $(c)$, and 4$s$ $(d)$.

The dependencies of the energy contribution to the binding energy of
magnetoexcitons Rydberg states for indirect magnetoexcitons on the magnetic
field and number of hBN layers for MoS2-hBN-MoS2 heterostructure are reported
in Fig. 5. For 1$s$ and 2$s$ states the energy contribution increases with the
increase of the external magnetic field and number of the hBN layers. Indirect
magnetoexcitons ionize in state 3$s$, when $B>11-15$ T, and in 4$s$ state,
when $B>5$ T. In Fig. 5$a$, it is shown the energy contribution for the $B$
exciton always falls between $\Delta E_{A_{\text{high}}}$ and $\Delta
E_{A_{\text{low}}}$: $\Delta E_{A_{\text{high}}}<\Delta E_{B}<\Delta
E_{A_{\text{low}}}$. The same holds for the states 2$s$, 3$s$ and 4$s$, but
not shown in Figs. 5$b$, 5$c$ and 5$d$. It is worth mentioning that Fig. 5.
shows a representative case for indirect magnetoexcitons in MoS2-hBN-MoS2
heterostructure. The same kind patterns and quantitative dependencies we
obtained for the MoSe2-hBN-MoSe2 heterostructures. The differences occur only
in the energy contribution magnitudes, which varies within up to 40% for
MoSe2-hBN-MoSe2 for the maximum value and indirect magnetoexcitons do not
ionize in the 3$s$ state and are ionized at larger values of the magnetic
field in 4$s$ state. In Fig. 6 are presented results for the tangerine-based
WSe2-hBN-WSe2 heterostructure. The tangerine-based heterostructures have the
same kind of patterns as are observed for the molybdenum-based
heterostructures. As a representative case in Fig. 7, we present the
comparison of the energy contribution to the binding energy of indirect
magnetoexcitons for MoSe2-hBN-MoSe2 heterostructure obtained by solving the
Schrödinger equation with the $V_{RK}$ and $V_{C}$ potentials. The analysis
shows that the $V_{RK}$ potential gives higher contributions to the binding
energy than the Coulomb potential for all states, and these contributions
increase with the increase of the magnetic field. The same qualitative
patterns are observed in other heterostructures.

(a) 1$s$ |  (b) 2$s$
---|---
|
(c) 3$s$ |  (d) 4$s$
|
Figure 5: The energy contribution from the magnetic field to the binding energies of indirect magnetoexcitons in MoS2-hBN-MoS2 heterostructure for Rydberg states 1$s$ $(a)$, 2$s$ $(b)$, 3$s$ $(c)$, and 4$s$ $(d)$. Calculations are performed using the Rytova-Keldysh potential. (a) 1$s$ |  (b) 2$s$
---|---
|
(c) 3$s$ |  (d) 4$s$
|
Figure 6: The energy contribution from the magnetic field to the binding energies of indirect magnetoexcitons in WSe2-hBN-WSe2 heterostructure for Rydberg states 1$s$ $(a)$, 2$s$ $(b)$, 3$s$ $(c)$, and 4$s$ $(d)$, respectively. Calculations are performed using the Rytova-Keldysh potential. (a) 1$s$ |  (b) 2$s$
---|---
|
(c) 3$s$ |  (d) 4$s$
|
Figure 7: Comparison between energy contribution when Schrödinger equation is
solved with the RK and Coulomb potentials for indirect $A_{\text{low}}$
magnetoexcitons in MoSe2-hBN-MoSe2 heterostructure for Rydberg states 1$s$
$(a)$, 2$s$ $(b)$, 3$s$ $(c)$, and 4$s$ $(d)$, respectively.

###  III.3 Diamagnetic coefficients

The diamagnetic coefficients $\sigma_{A_{\text{high/low}}}$ for $A$ and
$\sigma_{B}$ for $B$ direct magnetoexcitons in encapsulated MX2 monolayers
obtained in the framework of our approach are reported in Ref. [74]. Here we
report DMCs for direct magnetoexcitons in FS MX2 monolayers and indirect
magnetoexcitons in FS MX2-MX2 bilayers and MX2-hBN-MX2 heterostructures.
The DMCs of direct magnetoexcitons in FS monolayers along with the DMCs for
magnetoexcitons in encapsulated WSe2, WS2, MoSe2, and MoS2 are presented in
Table 2. Calculations are performed using the $V_{RK}$ potential. The DMCs
$\sigma_{A_{\text{high}}}<\sigma_{B}<\sigma_{A_{\text{low}}}$ for all states.
For the 1$s$ state $\sigma_{\text{hBN}}>\sigma_{\text{FS}}$ and the ratio
$\sigma_{\text{hBN}}/\sigma_{\text{FS}}$ varies from 1.6 to 2.6, depending on
monolayer parameters. For the 2$s$ state this ratio significantly exceeds 2.6.
Therefore, the screening effect in encapsulated monolayers substantially
increases the value of DMCs.

Table 2: The diamagnetic coefficients $\sigma_{\text{FS}}$ of direct magnetoexcitons in FS MX2 monolayers along with DMCs $\sigma_{\text{hBN}}$ in encapsulated MX2 monolayers reported in Ref. [74]. DMCs are given in $\mu$eV/T2.  |  Exciton | 1$s$ | 2$s$ | 3$s$
---|---|---|---|---
| $\sigma_{\text{FS}}$ |  $\sigma_{\text{hBN}}$ [74] | $\sigma_{\text{hBN}}/\sigma_{\text{FS}}$ | $\sigma_{\text{FS}}$ |  $\sigma_{\text{hBN}}$ [74] | $\sigma_{\text{hBN}}/\sigma_{\text{FS}}$ | $\sigma_{\text{FS}}$
WSe2 |  $A_{\text{high}}$ |  0.09 |  0.16 |  1.78 |  0.89 |  2.58 |  2.90 |  3.00
|  $A_{\text{low}}$ |  0.34 |  0.66 |  1.94 |  3.34 |  10.01 |  3.00 |
|  $B$ |  0.28 |  0.57 |  2.04 |  2.79 |  8.84 |  3.17 |
WS2 |  $A_{\text{high}}$ |  0.11 |  0.21 |  1.91 |  1.10 |  3.56 |  3.24 |  3.81
|  $A_{\text{low}}$ |  0.29 |  0.60 |  2.07 |  2.94 |  9.65 |  3.28 |
|  $B$ |  0.28 |  0.61 |  2.18 |  2.82 |  9.84 |  3.45 |
MoSe2 |  $A_{\text{high}}$ |  0.08 |  0.13 |  1.63 |  0.74 |  1.96 |  2.65 |  2.44
|  $A_{\text{low}}$ |  0.11 |  0.18 |  1.64 |  1.02 |  2.76 |  2.71 |  3.38
|  $B$ |  0.09 |  0.17 |  1.89 |  0.85 |  2.55 |  3.00 |  2.83
MoS2 |  $A_{\text{high}}$ |  0.08 |  0.21 |  2.63 |  0.77 |  3.56 |  4.62 |  2.61
|  $A_{\text{low}}$ |  0.28 |  0.55 |  1.96 |  2.77 |  8.64 |  3.12 |
|  $B$ |  0.11 |  0.22 |  2.00 |  1.08 |  3.50 |  3.24 |  3.68
Table 3: The diamagnetic coefficients of indirect magnetoexcitons in FS MX2-MX2 bilayers. $\sigma$ is given in $\mu$eV/T2. |  Exciton | 1$s$ | 2$s$ | 3$s$ | 4$s$
---|---|---|---|---|---
| $\sigma_{RK}$ | $\sigma_{C}$ | $\sigma_{RK}/\sigma_{C}$ | $\sigma_{RK}$ | $\sigma_{C}$ | $\sigma_{RK}/\sigma_{C}$ | $\sigma_{RK}$ | $\sigma_{C}$ | $\sigma_{RK}/\sigma_{C}$ | $\sigma_{RK}$ | $\sigma_{C}$ | $\sigma_{RK}/\sigma_{C}$
WSe2 |  $A_{\text{high}}$ |  0.15 |  0.04 |  3.75 |  1.10 |  0.32 |  3.44 |  3.45 |  1.20 |  2.88 |  7.73 |  3.21 |  2.41
|  $A_{\text{low}}$ |  0.47 |  0.12 |  3.92 |  3.90 |  1.21 |  3.22 |  12.49 |  5.06 |  2.47 |  27.03 |  |
|  $B$ |  0.40 |  0.11 |  3.64 |  3.29 |  1.05 |  3.13 |  |  4.31 |  |  |  |
WS2 |  $A_{\text{high}}$ |  0.09 |  0.05 |  1.8 |  0.89 |  0.43 |  2.07 |  3.00 |  1.69 |  1.78 |  |  4.65 |
|  $A_{\text{low}}$ |  0.34 |  0.11 |  3.09 |  3.34 |  1.15 |  2.9 |  |  4.87 |  |  |  |
|  $B$ |  0.28 |  0.11 |  2.55 |  2.79 |  1.15 |  2.43 |  |  4.87 |  |  |  |
MoSe2 |  $A_{\text{high}}$ |  0.13 |  0.03 |  4.33 |  0.91 |  0.23 |  3.95 |  2.79 |  0.85 |  3.28 |  |  2.24 |
|  $A_{\text{low}}$ |  0.17 |  0.04 |  4.25 |  1.23 |  0.31 |  3.97 |  3.82 |  1.18 |  3.24 |  |  3.17 |
|  $B$ |  0.14 |  0.03 |  4.67 |  1.05 |  0.27 |  3.89 |  3.22 |  0.99 |  3.25 |  |  2.65 |
MoS2 |  $A_{\text{high}}$ |  0.13 |  0.03 |  4.33 |  0.95 |  0.28 |  3.39 |  3.00 |  1.05 |  2.86 |  |  2.82 |
|  $A_{\text{low}}$ |  0.39 |  0.10 |  3.9 |  3.23 |  0.99 |  3.26 |  |  4.14 |  |  |  |
|  $B$ |  0.17 |  0.05 |  3.4 |  1.31 |  0.39 |  3.36 |  4.19 |  1.52 |  2.76 |  |  4.16 |
Table 4: The diamagnetic coefficients of indirect magnetoexcitons in the molybdenum-based heterostructures. $\sigma$ is given in $\mu$eV/T2. | MoSe2-hBN-MoSe2 | MoS2-hBN-MoS2
---|---|---
State |  $N$ | $\sigma_{A_{\text{high}}}$ | $\sigma_{A_{\text{low}}}$ | $\sigma_{B}$ | $\sigma_{A_{\text{high}}}$ | $\sigma_{A_{\text{low}}}$ | $\sigma_{B}$
|  |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$
1$s$ |  1 |  0.17 |  0.08 |  0.23 |  0.11 |  0.20 |  0.09 |  0.19 |  0.10 |  0.67 |  0.37 |  0.22 |  0.14
|  2 |  0.23 |  0.14 |  0.30 |  0.19 |  0.26 |  0.16 |  0.25 |  0.17 |  0.85 |  0.58 |  0.35 |  0.24
|  3 |  0.29 |  0.21 |  0.28 |  0.27 |  0.33 |  0.24 |  0.33 |  0.25 |  1.04 |  0.81 |  0.45 |  0.35
|  4 |  0.35 |  0.28 |  0.47 |  0.36 |  0.40 |  0.32 |  0.41 |  0.34 |  1.25 |  1.04 |  0.55 |  0.46
|  5 |  0.42 |  0.35 |  0.56 |  0.46 |  0.48 |  0.39 |  0.49 |  0.43 |  1.47 |  1.27 |  0.66 |  0.57
|  6 |  0.50 |  0.43 |  0.65 |  0.55 |  0.56 |  0.48 |  0.58 |  0.52 |  1.69 |  1.51 |  0.77 |  0.69
2$s$ |  1 |  2.21 |  1.41 |  3.09 |  2.00 |  |  1.67 |  2.61 |  1.83 |  |  |  3.75 |  2.69
|  2 |  2.55 |  1.90 |  |  2.65 |  |  2.23 |  3.03 |  2.43 |  |  |  |
|  3 |  2.90 |  2.35 |  |  3.22 |  |  2.74 |  3.46 |  2.96 |  |  |  |
|  4 |  3.24 |  2.76 |  |  |  |  |  |  |  |  |  |
Table 5: The diamagnetic coefficients of indirect magnetoexcitons in the tungsten-based heterostructures. $\sigma$ is given in $\mu$eV/T2. | WSe2-hBN-WSe2 | WS2-hBN-WS2
---|---|---
State |  $N$ | $\sigma_{A_{\text{high}}}$ | $\sigma_{A_{\text{low}}}$ | $\sigma_{B}$ | $\sigma_{A_{\text{high}}}$ | $\sigma_{A_{\text{low}}}$ | $\sigma_{B}$
|  |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$ |  $V_{RK}$ |  $V_{C}$
1$s$ |  1 |  0.21 |  0.11 |  0.80 |  0.43 |  0.69 |  0.37 |  0.28 |  0.16 |  0.74 |  0.43 |  0.72 |  0.37
|  2 |  0.28 |  0.18 |  1.00 |  0.67 |  0.86 |  0.58 |  0.37 |  0.26 |  0.94 |  0.67 |  0.92 |  0.58
|  3 |  0.36 |  0.27 |  1.21 |  0.93 |  1.06 |  0.81 |  0.48 |  0.38 |  1.16 |  0.93 |  1.14 |  0.81
|  4 |  0.45 |  0.36 |  1.44 |  1.18 |  1.27 |  1.04 |  0.59 |  0.50 |  1.39 |  1.18 |  1.38 |  1.04
|  5 |  0.54 |  0.46 |  1.68 |  1.44 |  1.48 |  1.27 |  0.70 |  0.62 |  1.63 |  1.44 |  1.62 |  1.27
|  6 |  0.63 |  0.55 |  1.93 |  1.71 |  1.70 |  1.51 |  0.82 |  0.75 |  1.88 |  1.71 |  1.86 |  1.51
2$s$ |  1 |  2.92 |  2.00 |  |  |  |  |  4.04 |  3.00 |  |  |  |
|  2 |  3.36 |  2.65 |  |  |  |  |  |  3.86 |  |  |  |

Results of calculations of DMCs for molybdenum and tungsten-based
heterostructures are presented in Tables 4 and 5, respectively. We present
results for $\sigma_{A_{\text{high}}}$ and $\sigma_{A_{\text{low}}}$ for
indirect $A$ magnetoexcitons obtained with parameters for each material that
give the highest and lowest binding energies, $E_{A_{\text{high/low}}}$ for
$A$ excitons and $\sigma_{B}$ for $B$ magnetoexcitons, respectively. For each
type of indirect magnetoexciton, we report two sets of $\sigma$: one from
solution of the Schrödinger equation with $V_{RK}$ potential, another one from
solution of the Schrödinger equation using $V_{C}$ potential. The data
analysis of Tables 4 and 5 shows the following important features for both
molybdenum and tungsten-based heterostructures: i. the DMCs of the $A$ and $B$
magnetoexcitons increase when the number of hBN layers increase; ii. the DMCs
obtained using Rytova-Keldysh potential are always bigger than one obtained
for the Coulomb potential; iii. for the states 3$s$ and 4$s$
$E_{0}\sim|E(B)-E_{0}|$ and Eq. (8) cannot be applied to extract
$\sigma_{A_{\text{high/low}}}$ and $\sigma_{B}$ from data; iv. for both the RK
and Coulomb potentials
$\sigma_{A_{\text{high}}}<\sigma_{B}<\sigma_{A_{\text{low}}}$. The latter
indicates that DMCs are sensitive to the exciton reduced mass and
polarizability of material: the smaller reduced mass of exciton and larger
polarizability lead to the bigger DMC value. A distinct feature of molybdenum-
based heterostructures is that for the 2$s$ state the DMCs exist in the
heterostructure with up to four hBN layers, while in the tungsten-based
heterostructures, only up to two hBN layers. When the number of hBN layers is
more than some specific number, the linear dependence of $\Delta E$ on $B^{2}$
is invalidated in 2$s$, 3$s$, and 4$s$ states for both the Rytova-Keldysh and
Coulomb potentials.

##  IV Conclusion

In the present paper, we have studied energy contributions from the external
magnetic field to the binding energies of $A$ and $B$ magnetoexcitons in 1$s$,
2$s$, 3$s$, and 4$s$ Rydberg states in WSe2, WS2, MoSe2, and MoS2 freestanding
and encapsulated monolayers, bilayer MX2-MX2, and MX2-hBN-MX2
heterostructures. The first time diamagnetic coefficients for the bilayer
MX2-MX2 and MX2-hBN-MX2 heterostructures are calculated and reported. We
consider the additional degree of freedom to tailor the binding energies and
diamagnetic coefficients for the MX2-hBN-MX2 heterostructures by varying the
number of hBN sheets between TMDC layers. The study is performed for direct
and indirect $A$ and $B$ excitons. For the $A$ magnetoexcitons we used two
sets of parameters given in literature that provide the highest
($A_{\text{low}}$ exciton) and lowest ($A_{\text{high}}$ exciton) binding
energy for excitons. For indirect magnetoexcitons in bilayer and van der Waals
heterostructures the dependence of energy contribution from the external
magnetic field to the binding energies and DMCs are studied using Rytova-
Keldysh and Coulomb potentials.

Our study shows that the energy contributions $\Delta E$ for direct and
indirect magnetoexcitons and DMCs are sensitive to the reduced mass of exciton
and the screening distance $\rho_{0}$: the smaller reduced mass and larger
$\rho_{0}$ lead to the bigger $\Delta E$ and DMCs.

The energy contributions $\Delta E$ for direct magnetoexcitons increase with
the increase of the external magnetic field, and $\Delta E$ is more
significant for 2$s$, 3$s$ and 4$s$ states. The DMCs for direct
magnetoexcitons in encapsulated monolayers are bigger than in FS monolayers,
$\sigma_{\text{hBN}}>\sigma_{\text{FS}}$, and the ratio
$\sigma_{\text{hBN}}/\sigma_{\text{FS}}$ varies from 1.6 to 3.6 depending on
monolayer parameters and the state of magnetoexcitons. Therefore, the
screening effect in encapsulated monolayers significantly increases the value
of DMCs.

For our investigation, a two-particle Schrödinger equation for indirect
excitons is solved using the Rytova-Keldysh and Coulomb potentials to
calculate binding energies and diamagnetic coefficients. We demonstrate strong
sensitivity of the energy contributions from the external magnetic field on
the type of the potential: $\Delta E_{RK}>\Delta E_{C}$ and the difference
increases with the increase of the magnetic field. For the $B$ magnetoexcitons
for both potentials $\Delta E_{A_{\text{high}}}<\Delta E_{B}<\Delta
E_{A_{\text{low}}}$. The DMCs for MX2-hBN-MX2 heterostructures obtained using
Rytova-Keldysh potential are always bigger than one obtained for the Coulomb
potential and the DMCs of the $A$ and $B$ magnetoexcitons increase with the
increase of the number of hBN layers. However, when the number of hBN layers
more than three the difference becomes very small. Such a result is
understandable because of the convergence of $V_{RK}$ and $V_{C}$ potentials
with the increase the numbers of hBN layers. Also, a distinct feature of
molybdenum-based heterostructures is that DMCs exist for the heterostructure
with up to four hBN layers, while tungsten-based heterostructures only up to
two hBN layers due to magnetoexcitons dissociation.
In the considered range of the magnetic field, the direct and indirect
magnetoexcitons have stable 1$s$ and 2$s$ Rydberg states. The 3$s$ and 4$s$
states magnetoexcitons dissociate at some values of the magnetic field
depending on parameters of material. The 3$s$ and 4$s$ states indirect
magnetoexcitons bound by the Coulomb potential dissociate at a lower magnetic
field than magnetoexcitons bound by the RK potential.
Related to bilayer MX2-MX2: the magnetic field’s contribution to the
magnetoexcitons binding energies is systematically bigger for the FS bilayer
than for the FS MX2 monolayer for all states when the Rytova-Keldysh potential
is used in calculations. The same conclusion is extended to DMCs.
Finally, our results raise the possibility of controlling the binding energies
of direct and indirect magnitoexcitons in monolayer MX2, bilayer MX2-MX2 and
MX2-hBN-MX2 heterostructures using the external magnetic field and open the
additional degree of freedom to tailor the binding energies and diamagnetic
coefficients for the MX2-hBN-MX2 heterostructures by varying the number of hBN
sheets between TMDC layers.

Acknowledgments. This work is supported by the U.S. Department of Defense
under Gran No. W911NF1810433 and PSC-CUNY Award No. 62261-00 50.

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