# Electric-field-tuned binding energies of trions in silicene, germanene, and
stanene monolayers

Roman Ya. Kezerashvili1,2, Shalva M.Tsiklauri,3 and Anastasia Spiridonova1
1New York City College of Technology, The City University of New York, USA
2The Graduate School and University Center, The City University of New York,
USA
3Borough of Manhattan Community College, The City University of New York, USA

###### Abstract

We predict a formation of intravalley and intervalley controllable trions in
buckled 2D materials such as silicene, germanene and stanene monolayers in an
external electric field. A study is performed within the framework of a
nonrelativistic potential model using the method of hyperspherical harmonics
(HH). We solve the three-body Schrödinger equation with the Rytova-Keldysh
potential by expanding the wave functions of a trion in terms of the HH. A
resultant system of coupled differential equations are solved numerically.
Controllable ground state energies of intravalley and intervalley trions by
the external electric field are presented. The dependencies of the binding
energy (BE) of trions in the silicene, germanene and stanene as a function of
the electric field are qualitatively similar. BEs of trions formed by $A$ and
$B$ excitons have a non-negligible differences which slightly increase as the
electric field increases. It is demonstrated that trion BEs can be controlled
by the external electric field and the dielectric environment has a
significant effect on the trion BE.

##  I Introduction

The prediction of the existence of trions Lampert58  consisting from an
exciton and electron or hole known as negatively or positively charged
excitons (X∓) gave rise to many theoretical and experimental studies of trions
in bulk materials, quantum-well systems, and two-dimensional (2D) materials.
Atomically thin transition metal dichalcogenides (TMDCs) are a class of two-
dimensional (2D) materials with remarkable optical and electronic properties
Kormanyos ; RMP . Since 2013, when trions were observed in two-dimensional
MoS2 monolayers MoS23Heinz , trions are the subject of intense studies, both
experimentally and theoretically in monolayer transition metal dichalcogenides
(TMDCs). In the past decade, different experimental groups observed and
reported the signature of a trion in TMDCs monolayers.

Theoretical studies of trions have integrated a wide variety of techniques and
carried out to calculate the binding energies (BEs) of excitonic complexes in
monolayer TMDCs (see reviews NScRev2015 ; BekReichman ; Durnev2018 ;
Kezerashvili2019 ; Suris2022 ). The first calculations of the BEs of trions in
TMDC was performed using the variational method and then different approaches
were used in calculations such as the stochastic variational method with
correlated Gaussian basis, the diffusion quantum Monte Carlo and path-integral
Monte Carlo methods, a direct diagonalization of the three-particle
Hamiltonian within the Tamm–Dancoff approximation, the finite element method,
the hyperspherical harmonic method and three-body Faddeev equations in
configuration or momentum spaces which we are citing in a chronological order
Reichman2013 ; TimeDepdensity matrix functional theory ;
BerkelbachDifMonteCarlo ; DenFuncTheoryPIMC ; Saxena ; VargaNano2015 ; Ganchev
; VargaPRB2016 ; Szyniszewski ; Szyniszewski2 ; Deilmann2017 ; KezFew2017 ;
magnetic ; Chaves2018 ; Florian2018 ; FilikhinKez ; Torche2019 ; photolum ;
Fu2019 ; Varga2020 ; ZhumagulovTamm ; ZhumagulovTamm2 ; Glazov2020 ; model ;
Variational2021 ; Marsusi2022 ; Glazov2023 ; Frederico2023 ; Keztsik2024 .
Results for BEs of trions in TMDC monolayers yielded impressively accurate
results consistent with experimental data.

Another category of 2D semiconductors are the buckled 2D allotropes of
silicon, germanium, and tin, known as silicene, germanene, and stanene, and
collectively referred to as Xenes Molle2017 ; Zheng2020 . Experimental studies
revealed one of the most crucial differences between Xenes and graphene and
TMDC that Xenes monolayer is not a perfectly flat sheet, but instead is
slightly buckled Matthes2013a ; Molle2017 . As a result, this unique structure
of Xenes makes them sensitive to the external electric field applied
perpendicular to the monolayer allowing to open and control the band gap. The
tunable band gap of Xenes gives researchers, among other things, extraordinary
in situ control over binding energies and optical properties of excitons in
these materials.

In contrast to TMDCs, there is no extensive research on excitonic complexes in
Xenes monolayers. The reason is that the synthesis of Xenes monolayers has not
been as successful and extensive, as for example, TMDCs monolayers because
Xenes monolayers are unstable in the air Acun2013 ; Tao2015 . In contrast to
graphene, silicene and other Xenes monolayers do not occur in nature.
Nevertheless, silicene nanoribons were experimentally synthesized on a metal
substrate Aufray2010 ; Padova2010 . This opened the way for silicene,
germanene, and stanene monolayers being transferred on metal like Au
Drummond2012 ; Ezawa2 ; Davila2014 ; Mannix2017 ; Grazianetti2020  and
substrates such as MoS2, Ir, ZrB2 Mannix2017 ; Sante2019  and hexagonal boron
nitride (hBN) Li2013 ; Khan2017  and synthesized as freestanding monolayers
Saxena2016 . Working with a metallic substrate is easier. For example,
silicene grown on Ag (111) Ezawa2012sil ; Tao2015  and germanene synthesized
by dry deposition on Au (111) surface Davila2014  have been thoroughly
investigated. However, depositing Xene on a metal leads to a significant
alteration of properties of the Xene monolayer. Depending on a substrate,
properties of Xenes monolayers vary, see Refs. Molle2017 ; Li2013  for the
list of Xenes properties on different substrates. In opposition to deposition
on a metal, depositing Xenes on hBN is harder. However, Xene deposited on hBN
preserves its properties because Xene and hBN weakly interact Li2013 . Xenes
monolayers deposited on hBN present a particular interest for studying
magnetoexcitons in monolayer and vdWHs KezSpir2022Xenes .

Xenes optical and magneto-optical properties have been addressed in Refs.
Bechstedt2012 ; Stille2012 ; Matthes2013a ; Fadaie2016  and Muoi2020 ;
Chowdhury2016 , respectively. Different physical phenomena, such as the Hall
effect Ezawa2012sil , the valley-locked spin-dependent Seeback effect Zhai2018
, anomalous quantum Hall effect EzawaAPS2012 , quantum spin Hall effect
Zhao2020 , and the Landau levels Ezawa2012sil ; Tsaran2014 ; Muoi2020  are
studied because of their essential role in applications of Xenes monolayers in
nano- and quantum devices Chen2018 ; Lyu2019 ; Glavin2020 ; Tao2020 ; Zhao2020
.

Because of the band inversion, these honeycomb materials are also topological
insulators Bechstedt2021 ; Liu2011 ; Ezawa2015 ; Matthes2016 ; Matusalem2017 ;
Yu2017 . The existence of an excitonic insulator phase in silicene, germanene
and stanene have been first studied by Brunetti et al. BBKPL2019 ; BBKPRB2019
in the framework of the effective mass approximation. The influence of the
screening, band dispersion and external electric field on transitions in Xenes
between excitonic, topological, and trivial insulator phases were investigated
in Pulci2023 .

Currently, there is a shortage of research on exciton complexes in Xenes. In
particular, a study of the formation of three quasiparticle states, trions, in
Xenes monolayers. In this paper, we address this gap and focus on a
theoretical investigation of trions in Xenes within the method of
hyperspherical harmonics (HH).

The paper is organized as follows. In Sec. II, we present the theoretical
model and formalism to study Mott-Wannier trions in Xenes within the framework
of the method of hyperspherical harmonics. We consider a nonrelativistic
potential model for three interacting electrons and holes system and employ
the three-body Schrödinger equation in the effective mass approximation. In
the framework of the HH method, the Schrödinger equation with the Rytova-
Keldysh potential Ritova ; Keldysh  is reduced to the system of coupled
differential equations for the hyperradial functions. A numerical solution of
this system provides the BE and wavefunction for trions in Xenes. In Sec. III,
we discuss a classification of intravalley and intervalley trions in Xenes
monolayers (Subsec. III.1) and present results of calculations of controllable
ground state energies of intravalley and intervalley trions by the external
electric field (Subsec. III.2). Here we analyze the dependencies of the BEs
and probability distributions of three bound particles on the external
electric field. Based on the capability to control a BE and compactness of
trions in Xenes by the external electric field, we discuss a proposal of a
possible trions crystallization in Xenes and estimate corresponding densities
of composite fermions gas. Conclusions follow in Sec. IV.

##  II Theoretical model

We start by providing the framework of the low-energy model describing exciton
states in Xenes monolayers and heterostructures when the external electric
field perpendicular to a monolayer is present. Silicene, germanene, and
stanene monolayers are 2D buckled materials and have honeycomb structure like
graphene. However, in contrast to graphene, the most stable form of Si, Ge,
and Sn monolayers is honeycomb lattice with the offset of triangular
sublattices A and B with respect to the monolayer’s plane. The offset is
denoted by $d_{0}$ and known as the buckling constant or buckling factor. As a
result of the buckled structure of Xenes monolayers, they are sensitive to the
external electric field applied perpendicular to a monolayer. This manifests
as an on-site potential difference between sublattices when the electric field
is applied. When the external electric field is not applied, the band
structure of Xenes monolayers in the vicinity of the $K/K^{\prime}$ points
resembles graphene band structure. However, the intrinsic gaps of Xenes are
larger than the band gaps in graphene. The application of the external
electric field perpendicular to the monolayer leads to a potential difference
between sublattices A and B that changes the band gap resulting in changes in
the effective masses of the electrons and holes.

The single-particle spectrum of electronic states in monolayer Xenes in the
electric field acting along the $z$-axis in the vicinity of the $K/K^{\prime}$
points reads ($\hslash=c=1$) Tabert :

$\hat{H}=v_{F}\left(\xi
p_{x}\hat{\tau}_{x}+p_{y}\hat{\tau}_{y}\right)-\xi\Delta_{gap}\hat{\sigma}_{z}\hat{\tau}_{z}+\Delta_{z}\hat{\tau}_{z}.$
(1)

In Eq. (1) $v_{F}$ is the Fermi velocity, $\xi,\sigma=\pm 1$ are the valley
and spin indices, respectively, $p_{x}$ and $p_{y}$ are the components of
momentum in the $xy$-plane of the monolayer relative to the $K$ point,
$\hat{\tau}$ and $\hat{\sigma}$ are the pseudospin and real spin Pauli
matrices, respectively, $2\Delta_{gap}$ is the intrinsic band gap, and
$\Delta_{z}=ed_{0}E_{\perp}$ is the gap induced by the electric field,
$E_{\perp}$, acting along the $z$-axis. The first term in Hamiltonian (1) is
the same as that of the low-energy Hamiltonian in graphene CastroNeto2009a ;
Abergel2010 , the second term is the spin-orbit coupling with an intrinsic
band gap of $2\Delta_{gap}$, the last term describes the sublattice potential
difference appearing when the external electric field is applied Drummond2012
; Ezawa1 ; Ezawa2 ; Ezawa3 .

From Eq. (1), the low-energy eigenvalues for charge carriers near the
$K/K^{\prime}$ points can be written as Tabert :

$E(k)=\sqrt{\Delta_{\xi\sigma}^{2}+v_{F}^{2}p^{2}},$ (2)

where

$\Delta_{\xi\sigma}=|\xi\sigma\Delta_{gap}-ed_{0}E_{\perp}|$ (3)

is the electric-field-dependent band gap at $p=0$. When the there is no
external electric field, the spin-up and spin-down bands of the valence and
conduction bands are degenerate. Application of $E_{\perp}$ lifts this
degeneracy by splitting both the valence and conduction bands. When
$\xi=-\sigma$, the gap is large, since conduction and valence bands are the
furthest from the Fermi energy level, and the electron and hole form $A$
excitons. When $\xi=\sigma$, the gap is small, since conduction and valence
bands are the closest, and the electron and hole form $B$ exciton. To open the
band gap $\Delta_{\xi\sigma}$, a critical value of the electric field has to
be achieved $E_{c}=\Delta_{gap}/(ed_{0})$ and then the lower bands form a
Dirac cone at $K/K^{\prime}$ points. The critical values of the electric field
for Xenes monolayers are given in Ref. BBKPRB2019 . Parabolic conduction and
valence bands in the vicinity of $K/K^{\prime}$ points allow to find the
effective mass of charge carriers as $m=\Delta_{\xi\sigma}/v_{F}^{2}$. In
Xenes monolayers the lowest conduction and the highest valence bands are
symmetric, therefore, the effective masses of an electron and hole are the
same and can be written as a function of the external electric field:

$m=\frac{\lvert\xi\sigma\Delta_{gap}-ed_{0}E_{\perp}\rvert}{v_{F}^{2}}.$ (4)

Analyses of Eq. (4) shows that the value of the electron or hole effective
mass depends on the band gap, Fermi velocity, buckling constant and is a
function of $E_{\perp}$.

###  II.1 Effective mass approach for trions in buckled 2D materials

The promotion of an electron from the filled valence band to the empty
conduction band leaves an empty electron state in the valence band. The
description of such many-body system can be reduced to the two-particle
problem of the negatively charged conduction electron interacting with a
positively charged valence hole that form an exciton or other excitonic
complexes such as charged excitons or trions. The trions are formed when an
exciton binds another electron or hole to form a negatively or positively
charged three-particle system: $X^{-}$ or $X^{+}$, respectively. A description
of properties of trions requires a solution of a three-particle problem. In
buckled two-dimensional monolayers, the resulting trions we consider as the
Wannier-Mott trions since the correlation between an electron and a hole
extends over many lattice periods. The representation of the electron-hole
pair bound in a Wannier-Mott exciton shows the strong spatial correlation of
these two constituents. Therefore, we are assuming that the interaction of the
exciton with the third particle (electron or hole) leads to a large-radius
type system.

We follow the approach in which one assumes that the electron and hole bands
are isotropic and parabolic, which is a good approximation for the low-energy
spectrum of 2D materials. This form of the Hamiltonian implies that both the
electron and hole single particle states form a single parabolic band. The
corresponding eigenproblem equation reduces to the Schrödinger equation in the
effective mass approximation. This approach is common in the literature to
describe excitons and trions in 2D materials. See, for example, Refs.
Kormanyos ; BekReichman ; Kezerashvili2019 ; Keztsik2024 ; Berkelbach2013 ;
Fogler2014 ; Berman2016 ; Wang2018 ; Berman2017 . Below, we follow the
effective mass approximation.

Figure 1: (Color online) Schematics of partition trees of Jacobi coordinates
when two particles have the same masses. The electron and hole in one
electron-hole pair have the same masses. This corresponds to the intravalley
X- and X+ trions.

In order to obtain the eigenfunctions and eigenenergies of a 2D trion in Xenes
when the electric field is perpendicular to the Xenes monolayer, we write the
Schrödinger equation for an interacting three-particle electron-hole system.
Because we are considering the varying electric field $E_{\perp}$, which is
directed along $z$-axis, the corresponding term in the 2D Schrödinger equation
vanishes. However, the effect of the electric field action is present through
the effective mass as follows from Eq. (4). Thus, one can write 2D Schrödinger
equation for the interacting three-particle electron-hole system within the
effective mass approximation in the following form

$\left[-\frac{\hslash^{2}}{2}\overset{3}{\underset{i=1}{\sum}}\frac{1}{m_{i}}\nabla_{i}^{2}+\overset{3}{\underset{i<j}{\sum}}V_{ij}(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|)\right]\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3})=E\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3}),$
(5)

where $m_{i}$ is the effective mass of the electron or hole defined by Eq. (4)
and $\mathbf{r}_{i}$ is the $i$th particle position. We assume only two types
of charge carriers: electron and hole with the corresponding effective masses.
In Eq. (5), $V_{ij}(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|)$ is the
interaction potential between $q_{i\text{ }}$ and $q_{j}$ charges in a 2D
material that was first derived in Ref. Ritova  and was independently obtained
by Keldysh Keldysh . We refer to it as the Rytova-Keldysh (RK) potential. The
Rytova-Keldysh potential describes the Coulomb interaction screened by the
polarization of the electron orbitals in the 2D lattice and has the following
form

$V_{ij}(r)=\frac{\pi
kq_{i}q_{j}}{2\kappa\rho_{0}}\left[H_{0}(\frac{r}{\rho_{0}})-Y_{0}(\frac{r}{\rho_{0}})\right],$
(6)

where $r=\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|$ is the relative
coordinate between two charge carriers $q_{i}$ and $q_{j}$. In Eq. (6)
$k=9\times 10^{9}$ N$\cdot$m2/C2, $\kappa$ is the dielectric constant of the
environment that is defined as $\kappa=(\varepsilon_{1}+\varepsilon_{2})/2$,
where $\varepsilon_{1}$ and $\varepsilon_{2}$ are the dielectric constants of
two materials that the Xenes layer is surrounded by, $\rho_{0\text{ }}$ is the
screening length, which sets the boundary between two different behaviors of
the potential due to a nonlocal macroscopic screening, and
$H_{0}(\frac{r}{\rho_{0}})$ and $Y_{0}(\frac{r}{\rho_{0}})$ are the Struve
function and Bessel function of the second kind, respectively. The screening
length $\rho_{0}$ can be written as $\rho_{0}=(2\pi\chi_{2D})/(\kappa)$
Berkelbach2013 , where $\chi_{2D}$ is the 2D polarizability, which in turn is
given by $\chi_{2D}=l\varepsilon/4\pi$ Keldysh , where $\varepsilon$ is the
bulk dielectric constant of the Xene monolayer. For large distances $\
r>>\rho_{0\text{ }}$the potential has the three-dimensional bare Coulomb tail
$V_{ij}(r\mathbf{)=}\frac{kq_{i}q_{j}}{\epsilon r}$, while at very small
distances, smaller than the screening length $r<<\rho_{0\text{ }}$, it becomes
a logarithmic potential like a potential of a point charge in two dimensions:
$V_{ij}(r\mathbf{)=}\frac{kq_{i}q_{j}}{\epsilon\rho_{0\text{
}}}\left[\ln\left(\frac{r}{2\rho_{0}}\right)+\gamma\right]$, where $\gamma$ is
the Euler constant. Therefore, the potential (6) becomes the standard bare
Coulomb potential at $r>>\rho_{0\text{ }}$ and diverges logarithmically at
$r<<\rho_{0\text{ }}.$ A crossover between these two regimes takes place
around distance $\rho_{0}$. Thus, at small distances between charge carriers
the short-range interaction strength decreases, while the long-range
interaction strength is unaffected and is the bare three-dimensional Coulomb
potential. It is worth noting that in Ref. Rubio  a very good approximation to
the RK potential that is simpler to use, fairly precise in both limits and
remarkably accurate for all distances was introduced.

To obtain a solution of the Schrödinger equation (5) for the negatively and
positively charged trions, we use the method of hyperspherical harmonics (HH)
Avery . The main idea of this method is the expansion of the wave function of
the trion in terms of HH that are the eigenfunctions of the angular part of
the Laplace operator in the four-dimensional (4D) space. As the first step,
let us separate the center-of-mass (c.m.) and the relative motion of three
particles and introduce sets of mass-scaled Jacobi coordinates Avery ;
JibutiSh . There are three equivalent sets of Jacobi coordinates and there is
the orthogonal transformation between these sets FM ; FilKez2024 . For three
non identical particles that have different masses the mass-scaled Jacobi
coordinates for the partition $i$ read as follows Avery ; JibutiSh ; FM :

$\displaystyle\mathbf{x}_{i}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m_{j}m_{k}}{(m_{j}+m_{k})\mu}}(\mathbf{r}_{j}-\mathbf{r}_{k}),$
$\displaystyle\mathbf{y}_{i}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m_{i}\left(m_{j}+m_{k}\right)}{(m_{i}+m_{j}+m_{k})\mu}}\left(\frac{m_{j}\mathbf{r}_{j}+m_{k}\mathbf{r}_{k}}{m_{j}+m_{k}}-\mathbf{r}_{i}\right),\text{
\ }i\neq j\neq k=1,2,3,$ (7)

where

$\mu=\sqrt{\frac{m_{i}m_{j}m_{k}}{m_{i}+m_{j}+m_{k}}}\ \ $ (8)

is the three-particle effective mass. In Eqs. (7) the subscripts $i$, $j$, and
$k$ are a cyclic permutation of the particle numbers. Trees of Jacobi
coordinates for a three-particle system, when two particles have the same
masses, are shown in Fig. 1.

The transformation (7) allows the separation of c.m. and relative motions of
three particles with Hamiltonian (5), and the Schrödinger equation for the
relative motion of the three-body system reads

$\left[-\frac{\hbar^{2}}{2\mu}(\nabla_{x_{i}}^{2}+\nabla_{y_{i}}^{2})+V(x_{1})+V(x_{2})+V(x_{3})\right]\Psi(\mathbf{x}_{i},\mathbf{y}_{i})=E\Psi(\mathbf{x}_{i},\mathbf{y}_{i}).\
\ $ (9)

In Eq. (9), $V(x_{i})$ is the interaction potential between two particles at
the relative distance $x_{1}$, $x_{2}$, and $x_{3}$, respectively, where
$x_{i}$ is the modulus of the Jacobi vector $\mathbf{x}_{{i}}$ (7), and (9) is
written for any of set $i=1,2,3$ of the Jacobi coordinates (7). The orthogonal
transformation between three equivalent sets of the Jacobi coordinates
simplifies calculations of matrix elements involving $V(x_{i})$ potentials.

This method is presented in detail in Ref. Keztsik2024  and is briefly
outlined here. We introduce in the 4D space the hyperradius
$\rho=\sqrt{x_{i}^{2}+y_{i}^{2}}$ and a set of three angles
$\Omega_{i}\equiv(\alpha_{i},\varphi_{x_{i}},\varphi_{y_{i}}),$ where
$\varphi_{x_{i}}$ and $\varphi_{y_{i}}$ are the polar angles for the Jacobi
vectors $\mathbf{x}_{i}$ and $\mathbf{y}_{j},$ respectively, and $\alpha_{i}$
is an angle defined as $x_{i}=\rho\cos\alpha_{i},$ $y_{i}=\rho\sin\alpha_{i}$.
Next, we rewrite the Schrödinger equation (9) for the trion using
hyperspherical coordinates in the 4D configuration space Keztsik2024 . This
transformation allows to reduce the solution of the problem for the three
particles in the 2D configuration space to the motion of one particle in the
4D configuration space. Then we introduce the hyperspherical harmonics
$\Phi_{K\lambda}(\Omega)$ in the 4D configuration space, which are the
eigenfunctions of the angular part of the generalized Laplace operator
$\widehat{K}^{2}(\Omega_{i})$ in the 4D configuration space
$\widehat{K}^{2}(\Omega_{i})\Phi_{K\lambda}(\Omega)=K(K+2)\Phi_{K\lambda}(\Omega)$
Avery , where $K$ is a grand angular momentum. Here we are using the short-
hand notation $\lambda\equiv$ $\\{l_{x},l_{y},L,M\\},$ where $L$ is the total
orbital angular momentum of the trion, $M$ is its projection, and the grand
angular momentum $K=2n+l_{x}+l_{y}$, $l_{x}$, where $l_{y}$ are angular
momentum corresponding to $\mathbf{x}$ and $\mathbf{y}$ Jacobi coordinates,
respectively, and $n$ $\geqslant 0$ is an integer number.

The functions $\Phi_{K\lambda}(\Omega)$ present a complete set of orthonormal
basis, and one can expand the wave function of the trion
$\Psi(\rho,\Omega_{i})$ in terms of the HH $\Phi_{K\lambda}(\Omega)$ as

$\Psi(\rho,\Omega_{i})=\rho^{-3/2}\sum_{{}_{K\lambda}}u_{K\lambda}(\rho)\Phi_{K\lambda}(\Omega_{i}).\
$ (10)

In Eq. (10) $u_{K\lambda}(\rho)$ are the hyperradial functions and by
substituting (10) into the Schrödinger equation written in the hyperspherical
coordinated Keztsik2024 , one can separate the radial and angular variables
and get a set of coupled differential equations for the hyperradial functions
$u_{K\lambda}(\rho)$:

$\left[\frac{d^{2}}{d\rho^{2}}-\frac{(K+1)^{2}-1/4}{\rho^{2}}+\kappa^{2}\right]u_{K\lambda}(\rho)=\frac{2\mu}{\hbar^{2}}\sum_{{}_{K^{{}^{\prime}}\lambda^{{}^{\prime}}}}\mathcal{W}_{K\lambda
K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\rho)u_{K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\rho).$
(11)

In Eq. (11) $\kappa^{2}=2mB/\hslash^{2},$ where $B$ is trion BE. The coupling
effective potential energy $\mathcal{W}_{K\lambda
K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\rho)$ is

$\mathcal{W}_{K\lambda
K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\rho)=\int\Phi_{K\lambda}^{\ast}(\Omega_{i})\sum_{i<j}^{3}V_{ij}(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|)\Phi_{K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\Omega_{i})d\Omega_{i}.$
(12)

The coupling effective interaction (12) is defined via the RK potential (6).
Substituting (6) into Eq. (12), one obtains the matrix elements of the
effective potential energies. The method of calculations of the effective
potential energies is given in Keztsik2024 . Calculations of matrix elements
$\mathcal{W}_{K\lambda K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\rho)$ of the
two-body $V_{ij}(\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|)$ interactions in
the hyperspherical harmonics expansion method for a three-body system is
greatly simplified by using the HH basis states appropriate for the partition
corresponding to the interacting pair. Using the matrix elements
$\mathcal{W}_{K\lambda K^{{}^{\prime}}\lambda^{{}^{\prime}}}(\rho)$ in Eq.
(11), one can solve the system of coupled differential equations numerically.
Results of numerical solutions of (12) for trions in Xenes are presented in
the next section.

##  III Trions in Xenes

We apply the present theoretical approach for calculations of the trion BEs in
the following Xenes monolayers: silicene (Si), germanene (Ge), and stanene
(Sn). A schematic representation of a trion in freestanding, supported, and
encapsulated Xenes monolayers in the external electric field $E_{z}=E_{\perp}$
perpendicular to the Xene layer is given in Fig. 2. The form of the trion wave
function (10) is the most general, not restricted to any particular mass ratio
of electrons and holes, and describes the three-particle relative motion. The
splitting of the conduction and valence bands in Xenes due to spin–orbit
coupling at non-zero electric fields leads to the formation of $A$ and $B$
excitons in the larger or smaller band gaps, with corresponding larger or
smaller masses of the electron and hole. Two of the three particles
constituting a positive or a negative trion in the Xene monolayer have the
same masses due to the equity of the mass of electron and hole that form an
exciton. However, they are not identical because they have different charges.
An intervalley trion may have the same masses of three particles but they are
not identical due to the different spin and/or valley indices. Below we
consider trions formed by a singlet bright $A$ or $B$ exciton and an electron
(X-) or hole (X+). Trion can be in either a single or triplet state.

Figure 2: (Color online) Schematic representation of the X- trion in the
external electric field perpendicular to the Xene layer in ($a$) freestanding,
($b$) supported and ($c$) encapsulated Xenes monolayer. $x_{i}$ and $y_{i}$
are Jacobi coordinates for the partition $i$.

###  III.1 Intravalley and intervalley trions

In 2D monolayer of Xenes intravalley and intervalley trions can be formed. Let
us first consider intravalley trions. An interaction of bright the $A$ or $B$
exciton with another charged carrier in the same valley, which can either be
an electron or a hole, forms $X^{-}$ or $X^{+}$ intravalley trion. Schematics
in Fig. 3 show the possible formations of the $X^{-}$ (Figs. 3$a$ and 3$b$)
and $X^{+}$ (Figs. 3$c$ and 3$d$). As follows from Eq. (4), the electron and
hole effective masses in $A$ exciton are the same. The $B$ excitons are
composed of electron and hole which also have equal masses but smaller than
masses of the electron and hole in $A$ excitons. We denoted these masses as
$m_{A}$ and $m_{B}$, respectively. The intravalley $X^{-}$ and $X^{+}$ trions
have the same two particles masses. As follows from Eq. (8), the effective
masses of the $X^{-}$ and $X^{+}$ trions formed by $A$ or $B$ exciton are
$\mu_{A}=\sqrt{\frac{m_{B}m_{A}^{2}}{m_{B}+2m_{A}}}$ and
$\mu_{B}=\sqrt{\frac{m_{A}m_{B}^{2}}{m_{A}+2m_{B}}}$, respectively Keztsik2024
. Because $m_{A}>m_{B}$, it follows that $\mu_{A}>\mu_{B}$. Therefore, the
effective mass of the $X^{-}$ and $X^{+}$ trions formed by the $A$ exciton is
larger than mass of the $X^{-}$ and $X^{+}$ trions formed by the $B$ exciton.
Due to the proportionality of the BE of trions to the three-particle effective
mass $\mu$ Keztsik2024 , a BE of $X^{-}$ and $X^{+}$ trions formed by the $A$
exciton is larger than BE of trions formed by the $B$ exciton. In the ground
state, both intravalley trions formed by charge carriers from the same valley
are spin-singlet trions.

Intervalley $X^{-}$ trions result from the attraction between $K$ valley
bright $A$ excitons with an electron from $K^{\prime}$ valley or bright $B$
excitons from the $K$ valley with an electron from the $K^{\prime}$ valley.
The effective masses of the $X^{-}$ intervalley trions formed by bright $A$
excitons is always larger than the effective masses of trions formed by $B$
excitons. The corresponding spin-valley configurations are presented in Figs.
3$e$, $3f$, $3g$, and $3h$. We should mention that the intervalley $X^{+}$
trions have the same masses as the $X^{-}$ trions formed by the $K$ valley
bright $A$ and $B$ excitons due to the interaction with a hole from the
$K^{\prime}$ valley. While intravalley trions are composed two particle with
the same masses, intervalley trions can be a system of three particles with
the same masses (Figs. 3$e$ and $3h$) or a three-particle system with two
particles that have the same masses (Figs. 3$f$ and $3g$).

Figure 3: (Color online) Schematic representation of the low-energy band
structure for 2D Xenes material and formation of the intravalley $X^{-}$ and
$X^{+}$ trions and intervalley $X^{-}$ trions. ($a$) and ($b$), and ($c$) and
($d$) represent the intravalley $X^{-}$ and $X^{+}$, respectively. ($e$),
($f$), ($g$), and ($h$) correspond to the intervalley $X^{-}$ trions with the
decrease masses from the biggest (spin-valley configuration ($e$)) to the
smallest one (spin-valley configuration ($h$)). Intervalley $X^{+}$ trions
have the same masses as $X^{-}$.

###  III.2
Results of calculations

The intravalley $X^{-}$ and $X^{+}$ trions in Xenes monolayers have two
particles (the electron and hole) that have the same masses and the third
particle (the electron or hole) has a different mass as seen in Figs. 3$a-3d$.
Therefore, we have to deal with three non-identical particles because two
particles with the same masses have different charges. For intervalley trions
we have two cases: $X^{-}$ and $X^{+}$ trions formed by the electron and hole
with the same masses and the third particle has a different mass (spin-valley
configurations in Figs. 3$f$ and 3$g$) or all three particles have the same
masses (spin-valley configurations in Figs. 3$e$ and 3$h$). In the latter
case, while three particles have the same masses, two belong to different
valleys.

Table 1: Input parameters for calculations of BEs of trions in the freestanding (FS) and encapsulating Xene monolayers. The notations are the following: 2$\Delta_{gap}$ is the total gap between conduction and valence bands; $d_{0}$ is the buckling parameter; $v_{F}$ is the Fermi velocity; $l$ is the monolayer thickness; $\varepsilon$ is the dielectric constant of the Xenes monolayer; $\rho_{0}$ is the screening distance; $E_{c}$ is a critical electric field. 2D material | $\kappa$ | 2$\Delta_{gap}$, meV |  $d_{0}$, Å |  $v_{F}$, $\times 10^{5},$ m/s |  $l,$ nm | $\varepsilon$ |  $\rho_{0},$ nm |  $E_{c},$ V/Å
---|---|---|---|---|---|---|---|---
FS Si | 1 | 1.9 Balendhran  | 0.46 Ni2012  | 6.5 Balendhran  | 0.4 Tao2015  | 11.9 | 2.38 | 0.55
FS Ge | 1 | 33 Balendhran  | 0.676 Ni2012  | 6.2 Balendhran  | 0.45 | 16 | 4.23 | 0.3
FS Sn | 1 | 101 Balendhran  | 0.85 Balendhran  | 5.5 Balendhran  | 0.5 | 24 | 5.99 | 0.2
Si (hBN, type I) | 4.89 | 27 Li2013  | 0.46 Li2013  | 4.33 Li2013  | 0.333 Li2013  | 11.9 | 0.41 | none

Figure 4: (Color online) Dependencies of the BE of intravalley and intervalley
trions in silicene, germanene, and stanene on the applied electric filed.
Curves 1, 3, and 5 correspond to the trions formed by $A$ excitons coupling an
electron ($X^{-}$) or hole ($X^{+}$), curves 2, 4, and 6 correspond to the
trions formed by $B$ excitons coupling an electron ($X^{-}$) or hole
($X^{+}$). ($a$) Freestanding intravalley singlet state trions. ($b$)
Freestanding intervalley singlet state trions. ($c$) Freestanding intervalley
triplet state trions. ($d$) Intravalley singlet state trions in Xenes
monolayer supported on an SiO2 substrate (curve I, III, and V) and hBN
encapsulated (curves II, IV, and VI). The plots for FS Xenes are truncated for
the external electric field $E_{\perp}$ less than a critical field.

In calculations of the BEs of trions in Xenes monolayers, we use the RK
potential. We solve the system of coupled differential equations (11) for the
hyperradial functions $u_{K\lambda}^{L}(\rho)$ numerically. By solving the
system of equations (11), one finds the binding energy as well as the
corresponding hyperradial functions. The latter allows one to construct the
wave function (10). Numerical solution of the coupled differential equations
requires the control of the convergence of the BEs for trions with respect to
the grand angular momentum $K$ for each value of the external electric field.
The relative convergence of the BE is checked as $\Delta
B/B=\left[B(K+2)-B(K)\right]/B(K)$, where $B(K)$ is the BE for the given $K$.
The analysis of the results for the BEs at different values of the electric
field shows that the reasonable convergence is reached for $K_{\max}$ = 12, so
we limit our considerations to this value.

We considered the formation of trions in the singlet and triplet ground states
and computed the corresponding BEs. The input parameters for calculations of
BEs of trions in the freestanding, supported on SiO2 and hBN encapsulated Xene
monolayers are presented in Table 1. The formation of Wannier–Mott excitons
due to the electron-hole interaction via the RK potential in semiconducting
phases in Xenes monolayers occurs when the external electric field exceed some
critical value which is unique to each material BBKPL2019 . A value of the
critical electric field is slightly different for $A$ excitons and for $B$
excitons. Following _ab initio_ calculations Drummond2012  which determined
that the crystal structure of silicene becomes unstable around 2.6 V/Å, we
consider in our calculations electric fields up to 2.7 V/Å and study the
formations of trions in Xenes at the range of the external electric field from
the critical value up to 2.7 V/Å. Results of calculations of dependencies of
trions BE on the external electric field are presented in Fig. 4.

The results of our calculations for the BEs of intravalley trions in a singlet
state in freestanding Xenes monolayer are presented in Fig. 4$a$. The BE
increases for all materials as $E_{\perp}$ increases. In addition, in FS Si,
Ge, and Sn, we observe a non-negligible difference in the BE of trions formed
by $A$ and $B$ excitons. These differences slightly increase as the electric
field increases. The trion BEs for FS Ge and FS Sn are qualitatively similar
to FS Si, but they are smaller than freestanding silicene. The curves for FS
Ge and Sn qualitatively resemble that of FS silicene, but at 2.7 V/Å FS
germanene reaches a maximum trions BE of 24.8 (24.3) meV, and the maximum BE
for FS stanene is roughly 21.1 (20.5) meV, significantly smaller than for FS
silicene $-$ 30.1 (29.8) meV. In parentheses the BEs of trions formed by $B$
excitons are given. The percent differences between the trion BE of FS Si and
FS Ge and FS Si and FS Sn at the largest electric field considered, are 82%
(81%) and 70% (69%), respectively.

In Fig. 4$b$ the dependencies of the binding energies for intervalley singlet
state trions in spin-valley configurations shown in Figs. 4$e$ and $3h$ as
functions of external electric field are displayed. The corresponding BE are
about $2-4$% larger and qualitatively resemble that of the intravalley trions:
i. the increase of the BE as the external electric field increases; ii. BEs
for FS silicene, germanene, and stanene are qualitatively similar; iii. a non-
negligible difference in the BE of trions formed by $A$ and $B$ excitons.

Figure 5: (Color online) Dependence of the probability distribution of three
particles in freestanding silicene on hyperradius $\rho$ and applied electric
field for ($a$) intravalley and ($b$) intervalley $X^{-}$ trion.

While the intravalley X- and X+ trions must exist in the spin-singlet state,
whereas the intervalley trions can exist in either the singlet or triplet
states. In Fig. 4$b$ the dependencies of the binding energies for intervalley
$L=1$ and $S=3/2$ state trions as functions of the applied external electric
field are shown. The BEs of triplet state trions in the FS silicene and
germanene monolayers are about three times smaller than that for singlet state
trions. Our calculations show unbound triplet state trions in the FS stanene.
The decrease of the trion binding energy in the triplet state is due to the
centrifugal repulsion.

In Fig. 4$d$ the intravalley trion BEs in Xenes monolayer supported on a SiO2
and hBN encapsulated are presented. In our calculations, the role of the
substrate and the encapsulation by hBN is considered through the potential (6)
with $\kappa=(\varepsilon_{1}+\varepsilon_{2})/2$, where $\varepsilon_{1}$ and
$\varepsilon_{2}$ are the dielectric constants of two materials that the Xene
layer is surrounded by. The comparison of the BEs shows that the FS monolayers
exhibit by far the largest BEs, due to the much weaker dielectric screening
induced by the environment compared to Xenes supported on the SiO2 or hBN
encapsulated. Hence, the trion BE is the largest for the freestanding Xenes
and gets progressively smaller for Xene supported on SiO2, and is the smallest
for hBN encapsulated. Thus, the trions BE is extremely sensitive to the
dielectric environment.

We calculated a probability distribution for three particles that form a trion
in the Xene. In Fig. 5 interparticle radial probability distributions for
intravalley and intervalley trions in the silicene monolayer are shown. The
difference in the probability distribution is related to the difference of the
effective masses $\mu$ of intravalley and intervalley trions. The probability
distribution for the intravalley trion is computed for the spin-valley
configuration shown in Fig. 3$a$ and that for the intervalley trions is
calculated for the spin-valley configuration in Fig. 3$e$ with the highest
mass. The analysis of the dependence of the probability distributions of three
particles on the hyperradius $\rho$ and the external electric field leads to
the following conclusion: the increase of the external electric field gives
the increase of the trions BE and makes trions more compact since the greater
binding energy increases the trion formation probability.

##  IV Conclusion

In summary, we predict the existence of electrically-controlled trions in
Xenes monolayers. We have applied the hyperspherical harmonics method to the
calculation of BEs for the trion and predict formations of trions in
freestanding, supported, and encapsulated Xenes in an external electric field
perpendicular to monolayers. Results of BEs calculation for trions formed by
$A$ and $B$ excitons show a non-negligible differences in trions energies
which slightly increase as the electric field increases. The BEs of the
intravalley and intervalley trions in the singlet state can be tuned in the
range of $24-31$ meV for silicene, $21-26$ meV for germanene, and $14-20$ meV
for stanene by varying the external electric field from the critical value
that is specific for each material up to 2.7 V/Å. Let us note that the trion
binding energy in Xenes are the same order as in TMDC monolayers Keztsik2024 .
The dependence of the BE for the silicene, germanene and stanene as a function
of the electric field are qualitatively similar. Our findings pave a way
toward manipulating the trion BE by the external electric field.

The maximum BEs of intervalley triplet state trions is about 7.4 meV and 8.5
meV in silicene and germanene, respectively. In stanene the triplet state
trions are unbound. To understand the importance of the screened electron-hole
interaction in formation of trions in Xenes we computed the BEs for Xenes
supported on SiO2 and hBN encapsulated. It is demonstrated that the dielectric
environment has a significant effect on the trion BE by decreasing it.

Results of calculations of the probability distribution show the increase of a
compactness of trions with the increase of the electric field, since the
greater binding energy increases the trion formation probability.

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