# The exciton-polariton properties of hexagonal BN based microcavity and their
potential applications in BEC and superconductivity

Huaiyuan Yang  State Key Laboratory for Artificial Microstructure and
Mesoscopic Physics, Frontier Science Center for Nano-optoelectronics and
School of Physics, Peking University, Beijing 100871, P. R. China  Xinqiang
Wang  State Key Laboratory for Artificial Microstructure and Mesoscopic
Physics, Frontier Science Center for Nano-optoelectronics and School of
Physics, Peking University, Beijing 100871, P. R. China  Collaborative
Innovation Center of Quantum Matter, Peking University, Beijing 100871, P. R.
China  Xin-Zheng Li<EMAIL_ADDRESS>State Key Laboratory for Artificial
Microstructure and Mesoscopic Physics, Frontier Science Center for Nano-
optoelectronics and School of Physics, Peking University, Beijing 100871, P.
R. China  Interdisciplinary Institute of Light-Element Quantum Materials,
Research Center for Light-Elemt Advanced Materials, and Collaborative
Innovation Center of Quantum Matter, Peking University, Beijing 100871,
People’s Republic of China  Peking University Yangtze Delta Institute of
Optoelectronics, Nantong, Jiangsu 226010, People’s Republic of China

###### Abstract

Microcavity exciton-polaritons are two-dimensional bosonic quasiparticles
composed by excitons and photons. Using model Hamiltonian with parameters
generated from ab initio density-functional theory and Bethe-Salpeter Equation
calculations, we investigate the exciton and the exciton-polariton properties
of hexagnonal boron nitride (hBN) based microcavity. We show that hBN based
microcavities, including monolayer and all-dielectric ones, are promising in
optoelectronic applications. Room temperature exciton-polariton Bose-Einstein
Condensation can be achieved because of the large oscillator strength and
binding energy of the exciton, and the strong interaction between the exciton-
polaritons and the longitudinal optical phonons.

Based on this BEC state, exciton-polariton mediated superconducting device can
also be fabricated at a few tens of Kelvin using the microcavity structure
proposed by Laussy et al.

##  I INTRODUCTION

Excitons and photons can be strongly coupled in semiconductors, resulting in
exciton-polariton Deng  _et al._ (2010). Microcavity is a platform to
demonstrate the existence of this quasiparticle. In these structure, excitons
in the semiconductor quantum well (QW) couple with the cavity photon Kavokin
and Malpuech (2003). The first experimental report of such exciton-polariton
based on microcavity comes from Weisbuch et al, in a GaAlAs/GaAs QW system
Weisbuch  _et al._ (1992). Formed by two kinds of bosons, exciton-polaritons
are bosons by definition. Under suitable environment, they can go through
Bose-Einstein Condensation (BEC). The lifetime of such microcavity exciton-
polariton is typically short, in order of several picoseconds. Therefore,
equilibrium BEC of exciton-polaritons in microcavity is normally hard to be
achieved in experiment.

By pumping excitons continuously into the system and with the help of
polariton-polariton scattering and polariton-phonon scattering, this lifetime
problem of the exciton-polariton can be circumvented. One example of such
dynamical nonequilibrium BEC state of exciton-polariton exists in CdTe-based
microcavities Kasprzak  _et al._ (2006). The extremely small effective mass of
an exciton-polariton (typically $10^{-4}\sim 10^{-5}m_{0}$ with $m_{0}$ being
the free electron mass) means that such BEC state could occur at moderate
temperatures if the Rabi coupling is large Plumhof  _et al._ (2014); Das  _et
al._ (2013); Li  _et al._ (2013); Su  _et al._ (2017). This remarkably
increases the BEC temperature when compared to the conventional ultracold
atomic systems (typically $\sim$nK). After the establishment of exciton-
polariton BEC, there are macroscopic occupations in the lower polariton branch
(LP) $k=0$ state, and coherence exists. This gives rise to possibility of
fabricating high-quality laser without the requirement of population inversion
Imamog¯lu  _et al._ (1996); Deng  _et al._ (2003), which is probably one of
the most energy-consuming steps in conventional lasers.

Besides BEC, exciton-polariton is also relevant to the designing of
superconducting devices. In conventional Bardeen-Copper-Schrieffer (BCS)
theory Bardeen  _et al._ (1957), electron-phonon coupling plays an important
role in mediating the attractive interactions between electrons. Other
mechanisms of superconductivity, however, also exist. These mechanisms include
the magnon mediated Kargarian  _et al._ (2016); Gong  _et al._ (2017), photon
mediated Kohn and Luttinger (1965); Schlawin  _et al._ (2019), the exciton
mediated Allender  _et al._ (1973); Ginzburg (1976), and the exciton-polariton
mediated ones Laussy  _et al._ (2010); Skopelitis  _et al._ (2018) etc. By
constructing a structure in which the BEC state of exciton-polariton can
induce effective attractive interaction between electrons in the adjacent
layer, Laussy et al. proposed that the bogolons in exciton-polariton BEC could
induce superconductivity Laussy  _et al._ (2010). Similar to the generation of
the exciton-polariton based BEC state and the related fabrication of laser as
described in the former paragraph, creation of the exciton-polariton is the
first step. Large binding energy and strong oscillator strength of the exciton
are favorable factors.

In the past decade, two-dimensional (2D) materials have received much
attention due to their unique electronic and optical properties, applicable to
nanoelectronics and nano-optical devices Wang  _et al._ (2012); Xia  _et al._
(2014); Balendhran  _et al._ (2015); Nakamura (2015); Kecik  _et al._ (2018);
Prete  _et al._ (2020a); Palummo  _et al._ (2015). Because of the reduced
screening effects, the exciton binding energies are substantially larger and
the oscillator strength are normally stronger than their bulk correspondences.
This may result in more stable exciton-polaritons at moderate temperatures.
When the Rabi coupling is large, microcavity based on these 2D materials can
provide a better platform for the demonstration of the BEC state, the BEC-
based laser, and the exciton-polariton mediated superconductor than the
conventional GaAs, GaN, and CdTe systems. Among these 2D materials, hexagonal
boron nitride (hBN) is unique due to its exceptionally large exciton binding
energies ($\sim$2 eV for monolayer and $\sim$0.7 eV for bulk) Wirtz  _et al._
(2006); Cudazzo  _et al._ (2016); Galvani  _et al._ (2016); Shen  _et al._
(2020). The wide bandgap also means that when room temperature BEC is
achieved, deep ultraviolet laser is likely. This is a feature highly desired
in the lighting industry. Based on such considerations, we explore in this
manuscript the exciton-polariton related properties in 2D materials based
macrocavity, with a special emphasis on hBN. The aim is to provide preliminary
theoretical results, which could be helpful for future experimental studies of
exciton-polariton microcavities.

This paper is organized as follows. In Sec. II, we introduce the background
theory for the methods we used in simulating these systems.

In Sec. III, we modify and apply the theories to realistic problems. Then we
present our results in detail.

These results include the properties of the exciton-polariton branches, the
BEC state simulated by the semiclassical Boltzmann equation, and the
possibility for the superconducting microcavity structures. In Sec. IV, a
brief summary of the conclusions is given.

##  II THEORY AND METHODS

###  II.1 Exciton-polaritons in microcavity

In a microcavity(see Fig.1(a)(b) for a typical microcavity structure), photons
are strongly confined in the $z$ direction. The energy dispersion of the
photon reads:

$E_{\mathrm{cav}}=\frac{\hbar c}{n}\sqrt{k_{z}^{2}+k_{\|}^{2}}.$ (1)

Here $n$ is the refractive index, $k_{\|}$ is the momentum of the photon in
the $x-y$ plane, and $k_{z}=\frac{j\pi n}{L}$ ($j$=1, 2, 3…) is its discrete
momentum in the $z$ direction.

The $j=1$ mode is the lowest one, and the other mode is well seperated in
energy. Therefore, we only consider $j=1$ mode throughout this manuscript. The
exciton energy dispersion is approximated by the effective mass model as:

$E_{\text{exc}}=E_{\text{exc0}}+\frac{\hbar^{2}k_{\|}^{2}}{2m_{\text{exc}}},$
(2)

where $E_{\text{exc0}}$ is the exciton energy at the $\Gamma$ point, and
$m_{\text{exc}}$ is the effective mass of the exciton to be fitted.

Under the rotating wave approximation, the exciton-polariton Hamiltonian can
be described in a simple way as Deng  _et al._ (2010):

$\displaystyle\hat{H}_{\mathrm{pol}}=$
$\displaystyle\hat{H}_{\mathrm{cav}}+\hat{H}_{\mathrm{exc}}+\hat{H}_{I}$ (3)
$\displaystyle=$ $\displaystyle\sum
E_{\mathrm{cav}}\left(k_{\|},k_{z}\right)\hat{a}_{k_{\|}}^{\dagger}\hat{a}_{k_{\|}}+\sum
E_{\operatorname{exc}}\left(k_{\|}\right)\hat{b}_{k_{\|}}^{\dagger}\hat{b}_{k_{\|}}$
$\displaystyle+\sum
g\left(\hat{a}_{k_{\|}}^{\dagger}\hat{b}_{k_{\|}}+\hat{a}_{k_{\|}}\hat{b}_{k_{\|}}^{\dagger}\right).$

$g$ is the Rabi coupling, which is half of the Rabi splitting.

After the diagonalization, the polariton Hamiltonian becomes:

$\hat{H}_{\mathrm{pol}}=\sum
E_{\mathrm{LP}}\hat{c}_{\text{LP},k_{\|}}^{\dagger}\hat{c}_{\text{LP},k_{\|}}+\sum
E_{\mathrm{UP}}\hat{c}_{\text{UP},k_{\|}}^{\dagger}\hat{c}_{\text{UP},k_{\|}}.$
(4)

Here the polariton operators are defined by:

$\hat{c}_{\text{LP/UP},k_{\|}}=C_{\text{LP/UP},k_{\|}}\hat{a}_{k_{\|}}+X_{\text{LP/UP},k_{\|}}\hat{b}_{k_{\|}}.$
(5)

$X$ and $C$ are the Hopfield coefficients obtained from the diagonalization.
The lower polariton (LP) and upper polariton (UP) energy dispersion equals:

$E_{\mathrm{LP}/\mathrm{UP}}\left(k_{\|}\right)=\frac{1}{2}\left[E_{\mathrm{exc}}+E_{\mathrm{cav}}\pm\sqrt{4g^{2}+\left(E_{\mathrm{exc}}-E_{\mathrm{cav}}\right)^{2}}\right].$
(6)

The lifetime of the LP mode ($\tau_{\text{pol}}$) is determined byDeng  _et
al._ (2010):

$\frac{1}{\tau_{\text{pol}}}=\frac{|C|^{2}}{\tau_{\text{cav}}}+\frac{|X|^{2}}{\tau_{\text{exc}}}.$
(7)

They describe the factions for the component of the LP mode from the exciton
and from the cavity photon. $\tau_{\text{exc}}$ is the radiative lifetime of
the exciton. $\tau_{\text{cav}}$ is the lifetime of the cavity photon mode.

The effective mass of LP is given byDeng  _et al._ (2010):

$\frac{1}{m_{\mathrm{LP}}}=\frac{|C|^{2}}{m_{\mathrm{cav}}}+\frac{|X|^{2}}{m_{\mathrm{exc}}},$
(8)

where the cavity photon effective mass is $m_{\mathrm{cav}}=\frac{\hbar
nj\pi}{cL}\sim 10^{-5}m_{0}$. This leads to extremely small effective mass of
the LP mode, which is crucial for the high critical temperature of BEC based
on this exciton-polariton.

###  II.2 Excitonic properties from first principle

Excitons are neutrally charged bosons formed by electron-hole pairs. Bethe-
Salpeter equation (BSE) is needed to describe them. As many ab initio
softwares (such as YAMBO Sangalli  _et al._ (2019a) and BerkeleyGW Deslippe
_et al._ (2012)) are available, it is convenient to get the excitonic
properties of a realistic material nowadays. The BSE Hamiltonian reads:

$H_{\text{e}\text{e}^{\prime}\text{h}\text{h}^{\prime}}=\left(E_{\text{e}}-E_{\text{h}}\right)\delta_{\text{e}\text{h},\text{e}^{\prime}\text{h}^{\prime}}+\left(f_{\text{e}}-f_{\text{h}}\right)\Xi_{\text{e}\text{e}^{\prime}\text{h}\text{h}^{\prime}},$
(9)

with $E_{\text{e(h)}}$ being the quasiparticle energy of the electron (hole)
and $f_{\text{e(h)}}$ being the occupation number. $\Xi$ is the Bethe-Salpeter
kernel calculated using the Kohn-Sham energies corrected by a scissor operator
and Kohn-Sham orbitals, representing the interaction between an electron and a
hole within the electron-hole pair. After diagonalizing the BSE Hamiltonian,
we get the exciton eigenenergy $E_{\lambda}$, with $\lambda$ labelling the
excitonic states. This determines the value of $E_{\text{exc0}}$ for each
excitonic branch in Eq. 2. The difference between the bandgap $E_{\text{g}}$
and $E_{\lambda}$ is the exciton binding energy $E_{\text{b}}^{\lambda}$.

Following Ref. Palummo  _et al._ , 2015, we calculate the exciton radiative
lifetime from the BSE results. In 2D materials, its value for an exciton in
state $\lambda$ at zero temperature reads:

$\tau_{\mathrm{\lambda}}(0)=\frac{\hbar^{2}c}{8\pi
e^{2}E_{\mathrm{\lambda}}(0)}\frac{A_{\mathrm{uc}}}{\mu_{\mathrm{\lambda}}^{2}}.$
(10)

Here $\mu_{\mathrm{\lambda}}^{2}$ is the square modulus of the exciton
transition dipole per unit cell in the $x-y$ plane obtained from the BSE
calculation, $A_{\mathrm{uc}}$ is the area of the unit cell, and
$E_{\mathrm{\lambda}}(0)$ is the exciton energy with zero wavevector. Using
the effective mass approximation for the exciton dispersion Palummo  _et al._
(2015), the average radiative lifetime in state $\lambda$ at temperature $T$
is:

$\tau_{\mathrm{\lambda}}^{T}=\tau_{\mathrm{\lambda}}(0)\frac{3}{4}\left(\frac{2m^{\mathrm{\lambda}}_{\text{exc}}c^{2}}{E_{\mathrm{\lambda}}(0)^{2}}\right)k_{\mathrm{B}}T.$
(11)

$m^{\mathrm{\lambda}}_{\text{exc}}$ is the exciton effective mass and
$k_{\mathrm{B}}$ is the Boltzmann constant.

###  II.3 Exciton-polariton dispersion and Rabi coupling

The exciton-polariton dispersion was not considered from first principle until
recent years. In 2015, Vasilevskiy et al. presented a classical
electrodynamics method to get the equation of motion for the exciton-polariton
modes, using parameters generated by density-functional theory (DFT)
calculations Vasilevskiy  _et al._ (2015).

Parallel to this classical electrodynamics method, exciton-polariton
dispersion in microcavity can also be described by quantum mechanics, from
quantized vector potential and exciton statesSavona  _et al._ (1994); Latini
_et al._ (2019). In this work, we follow the route of Savona et al. in Ref.
Savona  _et al._ , 1994, but use parameters generated by first principle BSE
calculations.

The differences between monolayer and bulk microcavity are addressed by taking
limits for the width of the QW and that of the cavity. Assuming a QW with a
width of $L^{\prime}$ embedded in a cavity with a width of $L$, and the
excitons are uniformly confined, the equation of motion for the exciton-
polariton reads (Eqs. 20-22 in Ref. Savona  _et al._ , 1994):

$\begin{array}[]{c}E^{2}-E_{\text{exc}}^{2}=\gamma\frac{E^{2}}{\alpha^{3}}[k_{0}L^{\prime}\alpha+2\cot\left(k_{0}\frac{L}{2}\alpha\right)\sin^{2}\left(k_{0}\frac{L^{\prime}}{2}\alpha\right)\\\
-\sin\left(k_{0}L^{\prime}\alpha\right)].\end{array}$ (12)

Here, $k_{0}=nE_{\text{exc}}/c$, and
$\alpha=\sqrt{|E^{2}/E_{\text{exc}}^{2}-k_{\|}^{2}/k_{0}^{2}|}$. $\gamma$ is
the effective interaction constant, which is directly related to the exciton
dipole moment $\mu_{\mathrm{\lambda}}$ by:

$\gamma=\frac{8\pi|\mu_{\lambda}|^{2}}{\hbar
ncA_{\text{uc}}\left(k_{0}L^{\prime}\right)^{2}}.$ (13)

$E$ appears on both sides of Eq. 12, meaning that it should be solved self-
consistently for each $k_{\|}$. This leads to the energy dispersions of all
the exciton-polariton branches. For monolayer (bulk) microcavity, we choose
the limit of $L^{\prime}$ approaching zero ($L^{\prime}=L$) and modify Eq. 12
to get the dispersions in Sec. IIIA&B. We find that the dispersions match
perfertly with Eq.6. By fitting with Eq. 6, the Rabi coupling $g$ can be
obtained.

###  II.4 Exciton in screening environment

Screening plays an important role in describing the excitonic properties. The
large binding energy in 2D materials comes from the reduced dielectric
screening. In real applications of optical devices, however, 2D materials are
often sandwiched by dielectric materials. The dielectric screening is expected
to influence the excitonic properties. It is much too computationally
expensive to perform converged ab initio BSE calculations with this factor
taken into account. Therefore, we employed a 2D Wannier exciton model Haug and
Koch (2009).

In this model, the exciton Schrödinger equation reads:

$\left[-\hbar^{2}\nabla_{r}^{2}/2\mu-e^{2}w^{2\text{D}}(r)\right]\psi_{\mathrm{exc}}(r)=E_{\text{b}}\psi_{\mathrm{exc}}(r).$
(14)

The screened Coulomb potential in two dimensions can be described by the
Keldysh potential Keldysh (1979):

$w^{2\text{D}}(r)=\left[\mathcal{H}_{0}\left(\Sigma
r/2r_{0}\right)-\mathcal{Y}_{0}\left(\Sigma
r/2r_{0}\right)\right]/8\epsilon_{0}r_{0}.$ (15)

Here $\mathcal{H}_{0}$ ($\mathcal{Y}_{0}$) is the zeroth order Struve function
(Bessel function of the second kind). $\Sigma=\epsilon_{a}+\epsilon_{b}$ is
the sum of the relative dielectric constants of the materials in both sides.
The screening length is $r_{0}=2\pi\alpha_{\text{2D}}$ and
$\alpha_{\text{2D}}$ is the 2D polarizability:

$\alpha_{2\text{D}}=\frac{2e^{2}}{(2\pi)^{2}}\sum_{c,v}\int_{\mathbf{k}}\frac{\left|\left\langle
u_{c,\mathbf{k}}\left|\nabla_{\mathbf{k}}\right|u_{v,\mathbf{k}}\right\rangle\right|^{2}}{E_{c,\mathbf{k}}-E_{v,\mathbf{k}}}d^{2}\mathbf{k}.$
(16)

They can be obtained from first principle calculations. After solving Eq. 14,
we get the exciton binding energy $E_{\text{b}}$ and the exciton Bohr radius
$a_{\text{B}}$ ($a_{\text{B}}=\int
d^{2}\mathbf{r}\psi^{*}_{\text{exc}}(\mathbf{r})r\psi_{\text{exc}}(\mathbf{r})$).

###  II.5 Exciton-polariton BEC described by the semiclassical Boltzmann
equation

We use the method of Porras et al. to describe the nonequilibrium BEC in a
semiclassical manner Porras  _et al._ (2002). The LP branch is divided into a
lower polaritonic (lp) region and an exciton reservoir region. The Boltzmann
equation for the exciton-polariton population reads:

$\begin{array}[]{c}\frac{dN_{k}^{\text{lp}}}{dt}=W_{k}^{\text{in}}n_{\text{x}}^{2}\left(1+N_{k}^{\text{lp}}\right)-W_{k}^{\text{out}}n_{\text{x}}N_{k}^{\text{lp}}-\Gamma_{k}^{\text{lp}}N_{k}^{\text{lp}}\\\
\\\
\frac{dn_{\text{x}}}{dt}=-\frac{1}{S}\sum_{k}dg_{k}^{\text{lp}}\left[W_{k}^{\text{in}}n_{\text{x}}^{2}\left(1+N_{k}^{\text{lp}}\right)-W_{k}^{\text{out}}n_{\text{x}}N_{k}^{\text{lp}}\right]\\\
\quad-\Gamma_{x}n_{\text{x}}+p_{x},\end{array}$ (17)

where $N_{k}^{\text{lp}}$ and $n_{\text{x}}$ refer to the occupation number of
the lower polariton region and the density of the exciton reservoir.
$W_{k}^{\text{in}}$ and $W_{k}^{\text{out}}$ are the scattering rates.
$\Gamma_{k}^{\text{lp}}$ and $\Gamma_{\text{x}}$ are the loss rates. Eq. 17
together with the equations describing the energy relaxation processes
(details see Ref. Porras  _et al._ , 2002) leads to a complete set of
equations for the exciton-polariton dynamics. After propagating the
population, the reservoir density, and the reservoir energy density to their
stationary solution, we can get the steady state for each pump strength.

###  II.6 Exciton-polariton mediated superconductivity

In conventional BCS theory, electron-phonon coupling induces effective
attractive interaction between electrons. Following this idea, Laussy et al.
proposed an exciton-polariton mediated mechanism for superconductivity Laussy
_et al._ (2010). When BEC happens, the exciton-polariton density is
tremendously large, the dipoles of the excitons can induce effective
attractive interaction between the electrons in the neighboring region. Here
we investigate the superconductivity transition temperature of the neighboring
electron gas layer in this scenario. We neglect the electron-phonon
interaction and electron Coulomb interaction to highlight the exciton-
polariton mechanism.

The microcavity structure is shown in Fig. 1 (c). The distance between the 2D
electron gas QW and the exciton-polariton QW is $L$, which we set as several
nanometers. The reduced Hamiltonian after the Bogoliubov transformation and
the mean-field approximation are made reads:

$\begin{array}[]{c}H=\sum_{\mathbf{k}}E_{\mathrm{el}}(\mathbf{k})e_{\mathbf{k}}^{\dagger}e_{\mathbf{k}}+\sum_{\mathbf{k}}E_{\mathrm{bog}}(\mathbf{k})b_{\mathbf{k}}^{\dagger}b_{\mathbf{k}}\\\
+\sum_{\mathbf{k},\mathbf{q}}M(\mathbf{q})e_{\mathbf{k}}^{\dagger}e_{\mathbf{k}+\mathbf{q}}\left(b_{-\mathbf{q}}^{\dagger}+b_{\mathbf{q}}\right).\end{array}$
(18)

$E_{\text{el}}(\mathbf{k})$ and
$E_{\mathrm{bog}}(\mathbf{k})=\sqrt{\tilde{E}_{\mathrm{pol}}(\mathbf{k})\left(\tilde{E}_{\mathrm{pol}}(\mathbf{k})+2UN_{0}A\right)}$
($\tilde{E}_{\mathrm{pol}}(\mathbf{k})\equiv
E_{\mathrm{pol}}(\mathbf{k})-E_{\mathrm{pol}}(\mathbf{0})$) are the in-plane
dispersion of the electrons and the bogolons. $N_{0}$ is the density of the
condensed exciton-polaritons and $U=6a_{\text{B}}^{2}E_{\text{b}}X^{4}/A$ is
the polariton-polariton interaction matrix element. $A$ is the quantization
area, $X$ is the exciton Hopfield coefficient.

The bogolon interacts with the electron through its excitonic fraction and the
interaction strength is

$M(\mathbf{q})=\sqrt{N_{\text{C}}A}XV_{\mathrm{X}}(\mathbf{q})\sqrt{\frac{E_{\mathrm{bog}}(\mathbf{q})-\tilde{E}_{\mathrm{pol}}(\mathbf{q})}{2UN_{0}A-E_{\mathrm{bog}}(\mathbf{q})+\tilde{E}_{\mathrm{pol}}(\mathbf{q})}}\\\
.$ (19)

$V_{X}$ is the electron-exciton interaction matrix element:

$V_{\mathrm{X}}(q)=\frac{e^{-qL}}{2\epsilon
A}\left\\{\frac{e^{2}}{q\left[1+\left(\beta_{\text{e}}qa_{\text{B}}/2\right)^{2}\right]^{3/2}}-\frac{e^{2}}{q\left[1+\left(\beta_{\text{h}}qa_{\text{B}}/2\right)^{2}\right]^{3/2}}+\frac{ed\beta_{\text{e}}}{\left[1+\left(\beta_{\text{e}}qa_{\text{B}}/2\right)^{2}\right]^{3/2}}+\frac{ed\beta_{\text{h}}}{\left[1+\left(\beta_{\text{h}}qa_{\text{B}}/2\right)^{2}\right]^{3/2}}\right\\},$
(20)

where
$\beta_{\text{e(h)}}=m_{\text{e(h)}}/\left(m_{\text{e}}+m_{\text{h}}\right)$.
The first and second terms within the brace of Eq. 20 are negligible compared
to the third and forth dipolar ones. The dipole $ed$ can be achieved by
applying an electric field perpendicular to the 2D plane. As the previous work
in exciton-polariton mediated superconductivity Laussy  _et al._ (2010, 2012);
Cherotchenko  _et al._ (2016), the effective interaction between electrons can
be expressed by Fröhlich potential Fröhlich (1952):

$V_{\mathrm{eff}}(\mathbf{q},\omega)=\frac{2M(\mathbf{q})^{2}E_{\mathrm{bog}}(\mathbf{q})}{(\hbar\omega)^{2}-E_{\mathrm{bog}}(\mathbf{q})^{2}}.$
(21)

Averaging the interaction over the 2D Fermi surface of the electrons, we get
the effective electron-electron interaction:

$U_{0}(\omega)=\frac{AN(0)}{2\pi}\int_{0}^{2\pi}V_{\mathrm{eff}}(q,\omega)d\theta.$
(22)

Here $q$ equals $\sqrt{2k^{2}_{F}(1+\text{cos}\theta)}$, and
$N(0)=m_{\text{e}}/\left(\pi\hbar^{2}\right)$ is the electron density of
states at the Fermi surface. $U_{0}$ is calculated numerically and substituted
into the gap equation through

$\Delta(\xi,T)=-\int_{-\infty}^{+\infty}\frac{U_{0}\left(\xi-\xi^{\prime}\right)\Delta\left(\xi^{\prime},T\right)\tanh\left(E/2k_{\mathrm{B}}T\right)}{2E}d\xi^{\prime},$
(23)

where $E=\sqrt{\Delta\left(\xi^{\prime},T\right)+\xi^{\prime 2}}$. Eq. 23 can
be solved by iteration as long as the initial guess is rational. The
superconductivity can happen if the $\Delta(0,T)$ is non-zero.

These equations form the basis upon which superconductivity is discussed in
this work. We note, however, that they are only applicable in the adiabatic
limit (the characteristic bogolon energy $\omega_{\text{B}}$ is much smaller
than the Fermi energy $E_{\text{F}}$). The final results presented in the
manuscript is beyond this scenario. To cope with this problem, the defects of
the Fröhlich potential will be discussed in Sec. III.C.

##  III RESULTS

###  III.1 Monolayer hBN microcavity

The structure of the monolayer hBN based microcavity is shown in Fig. 1 (a).
The monolayer is placed in the middle of the cavity and the dielectric
materials fill the space between the distributed Bragg reflectors (DBRs). In
this cavity, $L^{\prime}$ in Eq. 12 approaches 0, leading to
$\frac{k_{0}L^{\prime}\alpha-\sin(k_{0}L^{\prime}\alpha)}{(k_{0}L^{\prime})^{2}}\rightarrow
0$ and
$\frac{\sin^{2}(k_{0}\frac{L^{\prime}}{2}\alpha)}{(k_{0}L^{\prime})^{2}}\rightarrow\frac{\alpha^{2}}{4}$.
This means that Eq. 12 becomes:

$\begin{array}[]{c}E^{2}-E_{\text{exc}}^{2}=\gamma^{\prime}\frac{E^{2}}{\alpha}\cot\left(k_{0}\frac{L}{2}\alpha\right),\end{array}$
(24)

where

$\gamma^{\prime}=\frac{4\pi|\mu_{\lambda}|^{2}}{\hbar ncA_{\text{uc}}}.$ (25)

We apply Eq. 24 to several typical 2D nitrides (BN, AlN, GaN and InN) and 2D
transition metal dichalcogenides (TMDs, including MoS2, MoSe2, WS2 and WSe2),
using existing ab initio excitonic properties from BSE calculations Prete  _et
al._ (2020b); Palummo  _et al._ (2015). Table I shows the Rabi coupling of the
lowest bright exciton in these materials when detuning is zero and the
refractive index is unity. 2D nitrides based microcavity have larger Rabi
coupling than 2D TMDs in general, and hBN has the largest Rabi coupling among
them. This indicates that hBN is a very good candidate material for the
exciton-polariton based properties to be investigated.

Figure 1:  Sketch for (a) monolayer hBN microcavity, (b) all-dielectric hBN microcavity, and (c) model structure for the exciton-polariton mediated superconductivity.  Table 1:  Rabi coupling of the lowest bright exciton in eight typical 2D TMDs and 2D nitrides calculated by Eq. 24 based on existing BSE results from Refs. Prete  _et al._ , 2020b and Palummo  _et al._ , 2015. Detuning is set to zero, and the refractive index $n$ is unity. The unit of Rabi coupling $g$ is meV.  Materials | $g$ | Materials | $g$
---|---|---|---
MoS2 | 20.8 | BN | 99.4
MoSe2 | 19.0 | AlN | 85.4
WS2 | 23.5 | GaN | 70.3
WSe2 | 20.1 | InN | 32.4

For a more detailed understanding of the excitonic properties of hBN, we
performed DFT calculations using QUANTUM ESPRESSO (QE) and then the BSE
calculations using YAMBO Giannozzi  _et al._ (2009); Marini  _et al._ (2009);
Sangalli  _et al._ (2019b). This subsection focusses on the monolayer. Local
density approximation (LDA) is used in describing the Kohn-Sham exchange-
correlation potential, along with a $16\times 16\times 1$ k-point mesh for
Brillouin-zone sampling and a kinetic energy cutoff of 80 Ry for the expansion
of the wavefunctions. The direct LDA bandgap is 4.62 eV at $K$ as shown in
Fig. 2 (a). In the BSE calculation, a scissor operator of 2.87 eV is applied
to the Kohn-Sham energies, and a denser k-point mesh of $36\times 36\times 1$
is used to converge the results. The transitions between all four valence
bands and the lowest four conduction bands are included. An exciton binding
energy of 2.02 eV is obtained. The lowest bright exciton is mainly contributed
by the transition around the $K$ point.

Figure 2:  LDA band structure of (a) monolayer hBN and (b) bulk hBN
microcavity. The distance between the red and blue dashed line indicate the
direct bandgap.

Optical absorption spectrum at the random-phase approximation (RPA) level and
at the BSE level are shown in Fig. 3 (a). The binding energy and the
absorption spectrum are similar to recent studies Ferreira  _et al._ (2019);
Prete  _et al._ (2020b). Compared to the RPA result, the BSE spectrum captures
the excitonic effects, which substantially redshift the optical peak and
increase the intensity of the absorption edge. The four lowest exciton states
(labeled as 1, 2, 3, and 4) have prominent oscillator strengthes (larger than
$10\%$ of the maximum value). The degenerate exciton states 1 and 2 (3 and 4)
are 1s (2p) states coming from $K$ and $K^{\prime}$ respectively. Their
detailed information, including the positions of the absorption energy, the
effective interaction constant, and the radiative lifetimes at zero and room
temperatures, are shown in Table II. Due to large oscillator strength, the
zero temperature radiative lifetimes of these four exciton states are very
short, of the $10^{1}\sim 10^{2}$ fs. Combined with the exciton effective mass
extracted from the ab initio calculations Sponza  _et al._ (2018), we
calculate the room temperature radiative lifetimes in the last column. They
are at the order of $10^{1}$ ps. The increase of the radiative lifetime at
finite temperature is mainly due to the thermal distribution of the excitonic
states, away from the edge.

Table 2:  Exciton energy, effective interaction constant, Rabi coupling at zero detuning, radiative lifetimes at 0 K ($\tau_{\lambda}(0)$) and average radiative lifetimes at room temperature ($\tau_{\lambda}^{\text{RT}}$) for the 1, 2, 3, and 4 exciton states in 2D hBN.  Index | E(eV) |  $\gamma^{\prime}$(a.u.) |  $g$(meV) |  $\tau_{\lambda}(0)$(fs) |  $\tau_{\lambda}^{\text{RT}}$(ps)
---|---|---|---|---|---
1 | 5.473 | 5.486 | 99.43 | 29.15 | 30.85
2 | 5.473 | 5.486 | 99.43 | 29.15 | 2.06
3 | 6.356 | 1.561 | 53.04 | 88.22 | 76.60
4 | 6.356 | 1.561 | 53.04 | 88.22 | 21.62

Rabi coupling at different photon wavelength of the lowest two exciton is
shown in Fig. 3 (b) when $n=1$. Its magnitude is large, which decreases only
by $\sim$10 meV as the photon wavelength increases.

The approximate expression for the Rabi coupling Savona (1997); Sarchi (2007),

$\begin{array}[]{c}\hbar g_{\lambda}\simeq\sqrt{\frac{2\pi
E_{\lambda}^{\text{exc}}\mu_{\lambda}^{2}}{n^{2}LA_{\text{uc}}}},\end{array}$
(26)

also matches well with the results of Eq. 24.

To quantify the coupling between the four excitonic states and the cavity
photon mode, we construct a Hamiltonian:

$\hat{H}_{\mathrm{pol}}=\left(\begin{array}[]{ccccc}E_{\text{cav}}&g_{1}&g_{2}&g_{3}&g_{4}\\\
g_{1}&E_{\text{exc1}}&0&0&0\\\ g_{2}&0&E_{\text{exc2}}&0&0\\\
g_{3}&0&0&E_{\text{exc3}}&0\\\ g_{4}&0&0&0&E_{\text{exc4}}\\\
\end{array}\right).$ (27)

Diagonalizing this Hamiltonian, we get five exciton-polariton modes (labeled
as p1, p2, p3, p4 and p5, shown in Fig. 3 (c)). As
$E_{\text{exc1}}=E_{\text{exc2}}$, $E_{\text{exc3}}=E_{\text{exc4}}$ and
$E_{\text{exc3}}-E_{\text{exc1}}\gg g$, the splitting between p1 and p3 (p3
and p5) equals $2\sqrt{g_{1}^{2}+g_{2}^{2}}$ ($2\sqrt{g_{3}^{2}+g_{4}^{2}}$).

Each of the five exciton-polariton modes is a linear combination of the
original four excitonic states and one cavity photon mode. The corresponding
Hopfield coefficients are labelled as $C$ for the photon mode and $X_{1}$,
$X_{2}$, $X_{3}$, $X_{4}$ for the excitonic ones. In Fig. 4, we show the
square modulus of these coefficients. As can be seen in Fig. 4 (b), the p2
exciton-polariton mode is comprised by the contributions from $X_{1}$ and
$X_{2}$, meaning that it is purely excitonic state. The same feature holds for
the p4 exciton-polariton mode as it is comprised purely by contributions from
$X_{3}$ and $X_{4}$ (Fig. 4 (d)). The p1, p3, and p5 exciton-polariton modes,
on the other hand, show strong features of coupling, especially in the region
when the dispersion of the cavity photon mode intersects with the exciton
energies. This is clearly seen if we compare the dispersion of the exciton-
polariton modes in Fig. 3 (c) with the analysis of the Hopfield coefficients
in Fig. 4. As $E_{\text{exc3}}-E_{\text{exc1}}\gg g$, p1 is composed by the
photon mode and the first degenerate exciton pair before the crossing near
$E_{\text{exc1}}$(1st crossing in Fig. 3 (c)) and it is a bare excitonic state
after that. p5 is dominated by the second degenerate exciton pair and photon
fraction before and after the crossing near $E_{\text{exc3}}$(2nd crossing in
Fig. 3 (c)). p3 is a mixture of all five components as it goes through both
coupling regions. It is composed purely by the second degenerate exciton pair
after the 2nd crossing.

The lifetimes of these five exciton-polariton modes are evaluated by:

$\begin{array}[]{c}\frac{1}{\tau_{\text{pol}}}=\frac{|C|^{2}}{\tau_{\text{cav}}}+\frac{|X_{1}|^{2}}{\tau_{\text{exc1}}}+\frac{|X_{2}|^{2}}{\tau_{\text{exc2}}}+\frac{|X_{3}|^{2}}{\tau_{\text{exc3}}}+\frac{|X_{4}|^{2}}{\tau_{\text{exc4}}}.\end{array}$
(28)

We assume a typical cavity photon lifetime of 5 ps for high quality
microcavity. The results are shown in Fig. 3 (d), which can also be understood
by analyzing the composition of each exciton-polariton mode in Fig. 4. The
lifetimes of the p2 (p4) exciton-polariton mode is 3.6 ps (31.6 ps). They are
constant with respect to $k_{\parallel}$ in Fig. 3 (d), as they compositions
donot change with $k_{\parallel}$ in Fig. 4. As the lifetimes of the exciton
state 2 in and the photon mode (5 ps) are short compared with the others, the
exciton-polariton modes with large fraction of them have short lifetimes. This
leads to short lifetimes of the p1, p2, p3 modes before the 2nd crossing and
that of the p5 mode after it.

In reality, the dielectric materials filling the empty space in Fig.1(a)
induce screening. Here we quantify the screening effects on the exciton
binding energy and Rabi coupling using Eqs. 14 and 15. The screening length
$r_{0}$ is adjusted to match the ab initio binding energy in vacuum. The
resulting binding energy and the exciton Bohr radius as a function of the
dielectric constant is shown in Fig. 3 (e). The binding energy is
substantially reduced by the environmental screening and the radius increases
linearly with the dielectric constant. Under moderate temperatures, the
exciton is still stable with large dielectric constant. Based on such
excitonic properties after screening, we evaluate the Rabi coupling using Eq.
24. The exciton oscillator strength can be approximated by:

$|\mu_{\lambda}|^{2}\propto\frac{\left|p_{0}\right|^{2}\left|\psi_{\lambda}(0)\right|^{2}}{E_{\lambda}}.$
(29)

Here $p_{0}$ is the coupling strength, which we assume as constant.
$E_{\lambda}$ is the exciton energy, which equals $E_{\text{g}}-E_{\text{b}}$.
$E_{\text{g}}$ is approximated by $E_{\text{g}}=1.14E_{\text{b}}+5.07$, which
is fitted from ab initio resultsGuo  _et al._ (2021). $\psi_{\lambda}(0)$ is
the exciton wavefunction at zero relative distance between the electron and
the hole within the pair. The Rabi coupling decreases with increasing
dielectric constant as shown in Fig. 3 (f). Under moderate dielectric
screening, it is still larger than the room temperature thermal energy,
indicating the stability the exciton-polaritons.

Figure 3:  Exciton and polariton properties for monolayer hBN micrcavity. (a)
Absorption spectrum of 2D hBN calculated by BSE (black line) and RPA (red
line). (b) Rabi coupling of polaritons composited by cavity photon mode and
exciton 1(black line) for different photon wavelength using Eq. 24. The
corresponding results using approximate expression Eq. 26 are in red line. (c)
Dispersion of the five branches of polaritons(labels are in (d)) and cavity
photon (black solid line) when zero detuning. (d) Polariton lifetimes for the
five branches of polaritons. (e) Binding energy(black solid line) and radius
of the exciton(red dashed line) under different dielectric screening
environment. (f) Rabi coupling of polaritons composited by cavity photon mode
and exciton 1 under different dielectric screening environment.  Figure 4:
(a)-(e)Non-zero Hopfield coefficients for monolayer hBN microcavity polaritons
at zero detuning for the first polariton mode to the fifth polariton mode.

The most attractive property of such exciton-polariton is the possibility of
room temperature BEC. Using semiclassical Boltzmann equation, plenty of works
have been reported on the possibility and mechanism that drive the microcavity
exciton-polaritons to the lowest state ($k_{\|}$ state in the LP) to achieve
BEC Tassone  _et al._ (1997); Malpuech  _et al._ (2002); Porras  _et al._
(2002); Doan  _et al._ (2005); Hartwell and Snoke (2010). It has been proven
that polariton-polariton interaction and polariton-phonon interaction are
essential in the relaxation process. In conventional semiconductors such as
GaAs or CdTe, the Rabi coupling is so small that the optical phonons cannot
participate in this process and only acoustic phonons contribute Tassone  _et
al._ (1997); Malpuech  _et al._ (2002); Hartwell and Snoke (2010). For the
relaxation process of exciton-polariton in 2D hBN microcavity, scattering with
accoustic phonon is negligible as acoustic phonon dispersions are too flat to
fulfill the energy and momentum conservation law. Optical phonons, on the
other hand, participate in this process.

To theoretically describe this process, we follow a two-step strategy. The
interaction between exciton-polaritons is derived first by taking into account
of the interaction between their original components. The Hamiltonian is first
written as:

$\begin{array}[]{c}H_{\text{I}}=\sum_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{q}}\frac{M_{\text{xx}}}{2S}(a_{\mathbf{k}_{1}}^{\dagger}a_{\mathbf{k}_{2}}^{\dagger}a_{\mathbf{k}_{1}+\mathbf{q}}a_{\mathbf{k}_{2}-\mathbf{q}}+b_{\mathbf{k}_{1}}^{\dagger}b_{\mathbf{k}_{2}}^{\dagger}b_{\mathbf{k}_{1}+\mathbf{q}}b_{\mathbf{k}_{2}-\mathbf{q}})\\\
+\frac{\sigma_{\text{sat}}}{S}(a_{\mathbf{k}_{1}}^{\dagger}a_{\mathbf{k}_{2}}^{\dagger}a_{\mathbf{k}_{1}+\mathbf{q}}c_{\mathbf{k}_{2}-\mathbf{q}}+b_{\mathbf{k}_{1}}^{\dagger}b_{\mathbf{k}_{2}}^{\dagger}b_{\mathbf{k}_{1}+\mathbf{q}}c_{\mathbf{k}_{2}-\mathbf{q}})+\text{H.C.}\end{array}$
(30)

We only consider p1, p2 and p3 exciton-polaritons. $a$ and $b$ are the
operators for the 1 and 2 excitonic states, and $c$ is the cavity photon
operator. $M_{\text{xx}}=6E_{\text{b}}a_{\text{B}}^{2}$ and
$\sigma_{\text{sat}}=3.6ga_{\text{B}}^{2}$ are exciton-exciton and exciton-
photon interaction strengthes respectively. Transforming $a$, $b$ and $c$ into
the exciton-polariton basis:

$\begin{array}[]{l}p_{\text{p}1,\mathbf{k}}=X^{\text{p}1}_{1,\mathbf{k}}a_{\mathbf{k}}+X^{\text{p}1}_{2,\mathbf{k}}b_{\mathbf{k}}+C^{\text{p}1}_{\mathbf{k}}c_{\mathbf{k}},\\\
p_{\text{p}2,\mathbf{k}}=X^{\text{p}2}_{1,\mathbf{k}}a_{\mathbf{k}}+X^{\text{p}2}_{2,\mathbf{k}}b_{\mathbf{k}}+C^{\text{p}2}_{\mathbf{k}}c_{\mathbf{k}},\\\
p_{\text{p}3,\mathbf{k}}=X^{\text{p}3}_{1,\mathbf{k}}a_{\mathbf{k}}+X^{\text{p}3}_{2,\mathbf{k}}b_{\mathbf{k}}+C^{\text{p}3}_{\mathbf{k}}c_{\mathbf{k}},\end{array}$
(31)

representing $H_{\text{I}}$ in terms of exciton-polariton operators and
neglecting the p2 and p3 branch, we can get the lowest polariton branch:

$\begin{array}[]{c}H_{\text{p}1}=\sum_{\mathbf{k}}\epsilon_{\text{p}1,\mathbf{k}}p_{\text{p}1,\mathbf{k}}^{\dagger}p_{\text{p}1,\mathbf{k}}\\\
+\sum_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}\frac{V^{\text{p}1-\text{p}1}_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}}{2S}p_{\text{p}1,\mathbf{k_{1}}}^{\dagger}p_{\text{p}1,\mathbf{k_{2}}}^{\dagger}p_{\text{p}1,\mathbf{k_{3}}}p_{\text{p}1,\mathbf{k_{4}}},\end{array}$
(32)

where

$\begin{array}[]{c}\frac{1}{2}V_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4}}^{\text{p}1-\text{p}1}=(\frac{1}{2}M_{\text{x
x}}X^{\text{p}1}_{1,\mathbf{k}_{1}}X^{\text{p}1}_{1,\mathbf{k}_{2}}X^{\text{p}1}_{1,\mathbf{k}_{3}}X^{\text{p}1}_{1,\mathbf{k}_{4}}\\\
+\frac{1}{2}M_{\text{x
x}}X^{\text{p}1}_{2,\mathbf{k}_{1}}X^{\text{p}1}_{2,\mathbf{k}_{2}}X^{\text{p}1}_{2,\mathbf{k}_{3}}X^{\text{p}1}_{2,\mathbf{k}_{4}}+\sigma_{\text{sat}}C_{\mathbf{k}_{1}}^{\text{p}1}X_{1,\mathbf{k}_{2}}^{\text{p}1}X_{1,\mathbf{k}_{3}}^{\text{p}1}X_{1,\mathbf{k}_{4}}^{\text{p}1}\\\
+\sigma_{\text{sat}}X_{1,\mathbf{k}_{1}}^{\text{p}1}X_{1,\mathbf{k}_{1}}^{\text{p}1}C_{\mathbf{k}_{3}}^{\text{p}1}X_{1,\mathbf{k}_{4}}^{\text{p}1}+\sigma_{\text{sat}}C_{\mathbf{k}_{1}}^{\text{p}1}X_{2,\mathbf{k}_{2}}^{\text{p}1}X_{2,\mathbf{k}_{3}}^{\text{p}1}X_{2,\mathbf{k}_{4}}^{\text{p}1}\\\
+\sigma_{\text{sat}}X_{2,\mathbf{k}_{1}}^{\text{p}1}X_{2,\mathbf{k}_{1}}^{\text{p}1}C_{\mathbf{k}_{3}}^{\text{p}1}X_{2,\mathbf{k}_{4}}^{\text{p}1})\delta_{\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{k}_{3}+\mathbf{k}_{4}}.\end{array}$
(33)

The coupling between the exciton, photon and phonon has received much
attention, and an elementary excitation named “phonoriton” was proposed to
describe a quasiparticle resulting from their interactions Latini  _et al._
(2021). We note, however, that the coupling between the exciton and photon is
much stronger in microcavity when the Rabi coupling is large. Here we carry
out our simulations in this scenario and investigate the exciton-polariton
relaxation process toward BEC. As will be demonstrated, the scattering between
exciton-polariton and phonon plays an important role. The interactions between
the LO phonons and the exciton-polariton were taken into account through:

$\displaystyle H_{\mathrm{p1}-\mathrm{ph}}=$
$\displaystyle\sum_{\mathbf{k},\mathbf{q}}M(|\vec{k},\vec{q}|)\times\left(c_{\text{LO},\mathbf{q}}-c_{\text{LO},-\mathbf{q},}^{\dagger}\right)p_{\mathrm{p1},\mathbf{k}+\mathbf{q}}^{\dagger}p_{\mathrm{p1},\mathbf{k}}.$
(34)

The interaction matrix element can be described by the Fröhlich model Frohlich
_et al._ (1950); Frohlich (1954); Hartwell and Snoke (2010); Trallero-Giner
_et al._ (2021) as:

$\begin{array}[]{c}M(\mathbf{k},\mathbf{q})=iX_{\mathbf{k}}X_{\mathbf{k^{\prime}}}\sqrt{\frac{2\pi
e^{2}\hbar\omega_{\text{LO}}}{\vec{q}_{\|}^{2}V}\left(\frac{1}{\epsilon_{\infty}}-\frac{1}{\epsilon_{0}}\right)}\left[I_{\text{e}}^{\|}(|\mathbf{q}|)-I_{\text{h}}^{\|}(|\mathbf{q}|)\right],\end{array}$
(35)

where

$I_{\text{e(h)}}^{\|}=\left[1+\left(\frac{m_{\text{e(h)}}}{2m_{\text{exc}}}\left|q_{\|}\right|a_{\text{B}}\right)^{2}\right]^{-3/2},$
(36)

and $X_{\mathbf{k}}$ and $X_{\mathbf{k^{\prime}}}$ are Hopfield coefficients.

After these treatments, we solve the Boltzmann equation using the method
described in Section.II.E with polariton-polariton scattering and polariton-LO
phonon scattering included. The results are shown in Fig. 5(a) and (b). The
pumping threshold is $\rm 1\times 10^{2}\mu m^{-2}ps^{-2}$ ($\rm 35\times
10^{2}\mu m^{-2}ps^{-2}$) respectively when polariton-LO phonon scattering is
(is not) considered.

Although polariton-polariton scattering alone is enough to achieve BEC,
polariton-LO phonon scattering plays an important role in the exciton-
polariton relaxation process. It lowers the pumping threshold by nearly two
orders of magnitude. The strong polariton-LO phonon scattering comes from the
large Fröhlich interaction matrix element in Eq.35. It is large when the
transfer momentum $q$ is small, which is the case in the exciton-polariton
relaxation process. The population in logarithm scale is shown in Fig. 5 (c).
As the pumping strength is higher, the population changes to Boltzmann
distribution (black line) at threshold, and to Bose-Einstein distribution (red
line) when the pumping strength is higher than the threshold.

One side effect for the use of pumping is that the energy of the exciton-
polariton mode will go through a renormalization. We can estimate this using
the generalized Gross-Pitaevskii (GP) equation Wouters and Carusotto (2007);
Su  _et al._ (2017):

$\begin{array}[]{l}i\hbar\frac{\mathrm{d}\psi}{\mathrm{d}t}=\left[E_{0}+\alpha|\psi|^{2}+g_{\mathrm{R}}n_{\mathrm{R}}+\frac{i}{2}\left(\hbar
rn_{\mathrm{R}}-\Gamma\right)\right]\psi,\\\
\frac{\mathrm{d}n_{\mathrm{R}}}{\mathrm{d}t}=P-\left(\Gamma_{\mathrm{R}}+r|\psi|^{2}\right)n_{\mathrm{R}}.\end{array}$
(37)

Here $\psi$ and $n_{\mathrm{R}}$ are the exciton-polariton field and the
reservoir density. $E_{0}$ is the bare exciton-polariton energy. $\alpha$ and
$g_{\mathrm{R}}$ are the polariton-polariton interaction and lower polariton-
reservoir exciton interaction strengthes. $\Gamma$ and $\Gamma_{\text{R}}$ are
the exciton-polariton decay rates. $r$ is the condensation rate (including
contributions from both polariton-polariton and polariton-phonon scattering),
and $P$ is the pumping rate. All of these parameters can be estimated from
each specific material. The threshold pumping rate is
$P_{0}=\frac{\Gamma\Gamma_{\text{R}}}{\hbar r}$. The exciton-polariton energy
shift above and below the threshold is evaluated by:

$\begin{array}[]{l}\Delta
E_{\text{below}}=\frac{g_{\text{R}}P}{\Gamma_{\text{R}}},\\\ \Delta
E_{\text{above}}=\frac{\hbar
P\alpha}{\Gamma}-\frac{\Gamma_{\text{R}}\alpha}{r}+\frac{g_{\text{R}}\Gamma}{\hbar
r}.\end{array}$ (38)

The result for the 2D hBN microcavity is shown in Fig. 5 (d). The slope is
smaller above the threshold than before. The blueshift is of the meV order.
These values are small and can be reasonably neglected in the Boltzmann
equation.

Figure 5:  Results of room temperature BEC analysis using Boltzmann equation
and generalized GP equation. The cavity photon wavelength is 230nm. (a)
Condensed fraction when the optical phonons are included. (b) Condensed
fraction without optical phonons. (c) Three typical population curve under
three different pump strength. (d) Energy blueshift of lowest polariton mode
under different pump strength.

###  III.2 All-dielectric hBN microcavity

Now we investigate the all-dielectric hBN microcavity. We first focus on the
optical properties of bulk hBN. The LDA ground state is obtained using a
$6\times 6\times 2$ k-point mesh, along with a kinetic cutoff of 110 Ry for
the expansion of the wavefunctions. The band structure is shown in Fig. 2 (b).
The direct bandgap within LDA is 4.51 eV, located at $H$. It is marked in the
figure by dashed lines. In BSE calculation we use a dense k-point mesh of
$18\times 18\times 6$ and take the highest three valence bands and lowest two
conduction bands as transition bands. A scissor operator of 2.31 eV is applied
to the Kohn-Sham energies. The lowest exciton is mainly comprised by the
transitons around $K$ with a binding energy of 0.76 eV. The absorption
spectrum is shown in Fig. 6 (a). We find that the RPA spectrum, which only
takes inter-band absorption into account, deviates a lot from the experimental
results. As a comparision, the BSE spectrum with 0.17 eV broadening matches
the experiment results much better, indicating that the excitonic effects is
essensial here. The refractive index calculated from BSE result by
$n(\omega)=\sqrt{\frac{\text{Re}\epsilon+|\epsilon|}{2}}$ is shown in Fig. 6
(b). Near the absorption peak, the refractive index also shows a peak. The
degenerate exciton pair at 6.06 eV have the strongest oscillator strength, and
we will only focus on them.

The structure of all-dielectric hBN microcavity is shown in Fig. 1 (b). The
bulk hBN fills all the space of the microcavity between DBRs. Using Eq. 12
with $L^{\prime}=L$ and diagonalizing the following Hamiltonian:

$\hat{H}_{\mathrm{pol}}=\left(\begin{array}[]{ccc}E_{\text{cav}}&g_{1}&g_{2}\\\
g_{1}&E_{\text{exc1}}&0\\\ g_{2}&0&E_{\text{exc2}}\\\ \end{array}\right),$
(39)

we get the exciton-polariton energy dispersion. The results are shown in Fig.
6 (c). Two degenerate excitons together with photon mode form three exciton-
polariton modes, labeled as p1, p2 and p3. The middle branch, p2, is a linear
combination of two excitons and the p1 and p3 branches are mixtures of two
excitons and one photon modes. The Rabi coupling is several times larger than
the room temperature thermal energy at different photon wavelengths, as
demonstrated in Fig. 6 (d). It minimum value at around 210 nm results from the
maximum of the refractive index, which comes from the bright excitons at 6.06
eV. In all-dielectric microcavity, there are more phonon modes scattering with
the exciton-polaritons. Intuitively, the dynamics toward BEC will be even
easier. However, the radiative lifetime of the excitons at finite temperature
in bulk hBN is hard to be estimated, as the exciton effective mass is ill-
defined Paleari (2019). Consequently, a pure theoretical simulation of the BEC
properties as that in monolayer is not presented in this manuscript. We note,
however, that the radiative lifetime is approximately 800 ps at room
temperature according to a recent experiment Cao  _et al._ (2013). This value
indicates that the decay of the exciton-polaritons in all-dielectric BN
microcavity is even slower and the critical pumping strength will be low.
Therefore, bulk hBN based microcavity is also promising in applications
related to exciton-polariton.

Figure 6:  Exciton and polariton properties for all-dielectric hBN micrcavity.
(a)Absorption spectrum of bulk hBN calculated by BSE (black solid line) and
RPA (blue solid line) compared with the experimental resultsTarrio and
Schnatterly (1989) (red dashed line). (b) Refractive index calculated from the
BSE reults. (c) Dispersion of the three branches of polaritons and cavity
photon (black line). (d) Rabi coupling of polaritons composited by cavity
photon mode and exciton 1(black line) for different photon wavelength using
Eq. 12. The corresponding results using approximate expression Eq. 26 are in
red line.

###  III.3 Exciton-polariton mediated superconductivity

Recently, several theoretical works have proposed that superconductivity can
be induced by effective attractive interactions between the electrons mediated
by exciton-polariton Laussy  _et al._ (2010); Skopelitis  _et al._ (2018); Sun
_et al._ (2021). Here we investigate possibilities of this using hBN based
microcavity. The structure is shown in Fig. 1 (c). The dielectric constant of
the vacuum space is chosen as 4. We set the condensed exciton-polariton
density to be a very large value, $N_{\text{c}}=1\times 10^{13}~{}$cm-2. Now
the characteristic bogolon energy $\omega_{\text{B}}$ is $\sim$150 meV. Under
an electric field perpendicular to the 2D layer, the condensated exciton-
polaritons can interact with the electrons in the 2D electron gas QW. We set
the distance between these QWs as 2 nm. These parameters are rather realistic.
For simplicity, we ignore the Coulomb interaction and electron-phonon coupling
in the 2DEG to highlight the influence of exciton-polariton.

As shown in Fig. 7, the superconducting transition temperature $T_{\text{C}}$
calculated by the gap equation using the Fröhlich potential (Eqs. 21-23) is
high, even up to the room temperature. This is not physical because the large
value of $T_{\text{C}}$ is a consequence of the Fröhlich potential used Cotleţ
_et al._ (2016) (Eq. 21). Although the singularities of this potential can be
removed by computing principal values numerically Laussy  _et al._ (2012),
they will still develop two large shoulders in Eq. 22 Laussy  _et al._ (2010).
This may lead to a serious overestimation of $T_{\text{C}}$ .

In order to overcome these defects of the Fröhlich potential, we apply the
McMillan formula McMillan (1968); Allen and Dynes (1975)

$T_{\text{C}}=\frac{\omega_{\text{B}}}{1.2}\exp\left[-\frac{1.04(1+\lambda)}{\lambda}\right],$
(40)

where $\lambda=-U_{0}(0)$ is the electron-bogolon coupling strength. As shown
in Fig. 7, $T_{\text{C}}$ is substantially reduced, to several tens of Kelvin.
Upon this, we also note that the McMillan formula is only applicable when the
the adiabatic approximation is valid and the Migdal therom holds Migdal
(1958), i.e. $\omega_{\text{B}}\ll E_{\text{F}}$. Here
$M\equiv\omega_{\text{B}}/E_{\text{F}}\in[0.5,1]$, and the adiabatic
approximation and Migdal therom may fail.

To check how this impacts on the final results, we further consider the vertex
correction and calculated $T_{\text{C}}$ by Grimaldi  _et al._ (1995):

$T_{\text{C}}=\frac{1.13\omega_{\text{B}}}{\sqrt{e}(1+M)}\exp\left[\frac{1}{2}\frac{M}{(1+M)}\right]\exp\left[-\frac{1+\lambda_{z}[1/(1+M)]}{\lambda_{\Delta}}\right].$
(41)

The effective coupling $\lambda_{z}$ and $\lambda_{\Delta}$ are related to
adiabatic coupling strength $\lambda$, and they are defined in Ref. Grimaldi
_et al._ , 1995 in details. From Fig. 7, we see that the vertex correction
further reduces $T_{\text{C}}$ compared to the McMillan formula. As
$E_{\text{F}}$ gets larger, their $T_{\text{C}}$s tend to converge. When the
$E_{\text{F}}$ is smaller than $\omega_{\text{B}}$, the above two formulas are
not applicable, and the $T_{\text{C}}$ should be calculated by the non-
adiabatic limit fomulaSadovskii (2020):

$T_{\mathrm{c}}\sim\frac{E_{\text{F}}}{1+\frac{E_{\text{F}}}{\omega_{B}}}\exp\left(-\frac{1}{\lambda_{\text{NA}}}\right).$
(42)

Here
$\lambda_{\text{NA}}=\frac{N(0)}{\pi}\int_{0}^{2\pi}d\theta\frac{\left|M_{q}\right|^{2}}{\omega_{q}}$
and
$q^{2}=2m\left(\omega_{q}+E_{\mathrm{F}}\right)+k_{\mathrm{F}}^{2}-2\sqrt{2m\left(\omega_{q}+E_{\mathrm{F}}\right)}k_{\mathrm{F}}\cos\theta$.
$T_{\text{C}}$ is strongly suppressed and the value computed by 42 is
vanishingly small. We note, however, that the situation in the system
considered belongs to the region when vertex correction is applicable
($M\equiv\omega_{\text{B}}/E_{\text{F}}\in[0.5,1]$). Therefore, we conclude
that $T_{\text{C}}$ can reach several tens of Kelvin. Such 2D material based
microcavities is promising for the fabrication of superconducting devices,
based on this scenario of exciton-polariton mediated superconductivity.

Figure 7:  Exciton-polariton mediated superconductivity in BN based
microcavity structure shown in Fig. 1 (c), when $N_{\text{c}}=1\times
10^{13}~{}$cm-2,$\omega_{\text{B}}=150$meV, $L=2$ nm, $d=1$ nm, and
$\epsilon=4$. The figure shows $T_{\text{C}}$ calculated by McMillan formula
(Eq. 40, black line), by vertex correction formula (Eq. 41, red line) and by
gap equation using Fröhlich potential (Eqs. 21-23, blue line) with different
2DEG Fermi energy $E_{\text{F}}$.

##  IV CONCLUSION

In summary, we studied the exciton-polariton properties in hBN based
microcavities. Using ab initio calculations (DFT+BSE), we obtain exciton
properties including absorption spectrum, exciton energy, exciton radiative
lifetimes for hBN. Based on these ab initio results, we investigate the
exciton-polariton dispersion, Rabi coupling, Hopfield coefficients, and
exciton-polariton lifetimes for hBN based microcavities. The oscillator
strength in both monolayer and bulk hBN is very large, leading to large Rabi
coupling. There are five non-negligible exciton-polariton modes in the
monolayer hBN and three in the bulk hBN. We analyzed the component fraction
and lifetime of each mode. With the help of the Boltzmann equation, we find
that room temperature exciton-polariton BEC can be achieved in hBN-based
microcavity owing to the large oscillator strength, binding energy, and the
strong polariton-LO phonon interactions. Superconductivity at a few tens of
Kelvin may also be induced by polariton-electron interaction, if the
microcavity structure is specially designed. Overall, we conclude that hBN
microcavities are very suitable platforms for studying the rich physics
associated with exciton-polaritons. And we hope this work can stimulate more
experimental/theoretical studies and the predictions presented in this
manuscript can be further tested.

###### Acknowledgements.

We acknowledge helpful discussions with Junren Shi and Shiwu Gao. The authors
are supported by the Beijing Natural Science Foundation under Grant No.
Z200004, the Strategic Priority Research Program of the Chinese Academy of
Sciences Grant No. XDB33010400, the National Basic Research Programs of China
under Grand Nos. 2016YFA0300900, the National Science Foundation of China
under Grant Nos 11774003, 11934003, and 11634001. The computational resources
were provided by the supercomputer center in Peking University, China.

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