Ultra‐Confined Phonon Polaritons and Strongly Coupled Microcavity Exciton Polaritons in Monolayer MoSi2N4 and WSi2N4

Abstract The 2D semiconductors are an ideal platform for exploration of bosonic fluids composed of coupled photons and collective excitations of atoms or excitons, primarily due to large excitonic binding energies and strong light‐matter interaction. Based on first‐principles calculations, it is demonstrated that the phonon polaritons formed by two infrared‐active phonon modes in monolayer MoSi2N4 and WSi2N4 possess ultra‐high confinement factors of around ≈105 and 103, surpassing those of conventional polaritonic thin‐film materials by two orders of magnitude. It is observed that the first bright exciton possesses a substantial binding energies of 750 and 740 meV in these two monolayers, with the radiative recombination lifetimes as long as 25 and 188 ns, and the Rabi splitting of the formed cavity‐exciton polaritons reaching 373 and 321 meV, respectively. The effective masses of the cavity exciton polaritons are approximately 10−5 m e , providing the potential for high‐temperature quantum condensation. The ultra‐confined and ultra‐low‐loss phonon polaritons, as well as strongly‐coupled cavity exciton polaritons with ultra‐small polaritonic effective masses in these two monolayers, offering the flexible control of light at the nanoscale, probably leading to practical applications in nanophotonics, meta‐optics, and quantum materials.


Introduction
The increase of electric field strength through trapping light at the nanoscale in optical microcavities results in strong light-−MoO 3 flakes and 500/1.6 ps in hBN flakes, respectively. [8]These confinement factors are comparable to those of graphene plasmon polaritons, but the lifetimes are nearly one order of magnitude larger than that of graphene plasmon polaritons.However, unlike the traditional phonon polaritons in 3D polar materials that arise from the splitting of longitudinal optical (LO) and transverse optical (TO) phonon modes, when the dimensionality crosses over to two dimensions, the screened macroscopic Coulomb interaction for 2D systems becomes inversely proportional to in-plane wavevectors.This results in the absence of the LO-TO splitting at the Γ point. [9]Therefore, strictly speaking, the so-called phonon polaritons cannot be well-defined in polar monolayer materials. [10]Arising from the fact that 2D LO phonon modes in monolayer materials are associated with ultralow-loss and highly confined electromagnetic modes, the 2D LO phonon can be thus regarded as the 2D phonon polaritons, with properties similar to their 3D counterparts. [11]y Coulomb interactions, excitons are bound states with the pairing of electrons and holes, which play an important role in determining the optical properties in semiconducting materials.14] Furthermore, benefiting from the tunability of semiconductor microcavity, [15] the coupling strength and other physical properties of microcavity exciton polaritons can be significantly improved.For some 2D systems placed in microcavities with strong light-matter couplings, the Rabi splitting of exciton-polaritons can be enhanced to 200 -300 meV. [16,17]ere, we demonstrate the phonon and exciton polaritons of novel materials family monolayer MoSi 2 N 4 and WSi 2 N 4 .We first examine the structural stability, electronic, transport, phonon vibrations, and optical properties.Our results show that the quasiparticle (QP) bandgaps are renormalized to 3.20 and 3.03 eV considering spin-orbit coupling (SOC) effect for monolayer MoSi 2 N 4 and WSi 2 N 4 .We conduct an analysis of electron eigenvalues at the Γ point, considering factors such as atomic energy levels, crystal-field splitting, and the impact of SOC effects.We observe a notably large confinement factor of phonon polaritons formed by two infrared-active phonon modes in both monolayer MoSi 2 N 4 and WSi 2 N 4 .We find that the excitons in two monolayers possess considerable large binding energies of 0.75 and 0.74 eV, respectively.By utilizing many-body perturbation theory (MBPT), we explore exciton dynamics mechanisms, including electronelectron scatterings, electron-phonon scatterings, and intrinsic direct recombination.Theoretical calculations show that the Rabi splitting of monolayer MoSi 2 N 4 and WSi 2 N 4 reaches 373 and 321 meV for the first bright exciton, respectively.

Crystal Structures and Bandstructures
The crystal structures of monolayer MoSi 2 N 4 and WSi 2 N 4 are shown in Figure 1a-c, which possess seven sub-layers, belonging to the space group of P-6m2 (No. 187) (point group D 3h ).The optimized lattice constants of monolayer MoSi 2 N 4 are a = b = 2.89 Å, while those of WSi 2 N 4 are a = b = 2.90 Å, as listed in Table 1, which are in good agreement with the previous work. [18]These two structures both possess an in-plane mirror symmetry.
The calculated phonon dispersions with average phonon occupancy at 300 K described by the Bose-Einstein statistics, with the consideration of the 2D implementation of the nonanalyticalterm correction [9] for monolayer MoSi 2 N 4 and WSi 2 N 4 are shown in Figure 1d,h, which reveals that both phonon dispersions possess no imaginary frequencies, indicating that these two monolayers are thermally stable at low temperatures, and the optical phonon modes belonging to the twelve optical branches with largest frequencies are not well excited at room temperature in both monolayers.As shown in Figure 1d,h, two phonon bandgaps are observed up to 2.26 and 2.05 THz for monolayer MoSi 2 N 4 , and 2.50 and 1.96 THz for monolayer WSi 2 N 4 .By enforcing the rotational sum rules to the second force constants implemented in the Hiphive package, [19] the calculated dispersion for ZA modes around the Γ point manifests quadratic shape, and those for TA and LA modes are linear.
Here to eliminate the spurious polarization field from the neighboring monolayer, which may influence the atomic movements within the monolayer, the Coulomb cutoff for the macroscopic dielectric screening is used here.Generally the relationship between LO and TO phonon frequencies influenced by the Coulomb screening effects can be written as, [9,10] where Ω is the unit-cell volume, q is the unit vector of phonon momentum q, and Z * n is the Born effective charge tensor of each atom in the unit-cell. n = ê,n ∕ √ M n is the atom displacement along q direction, and ê,n is the phonon eigenvector, M n is the mass of n th atom.W(q) is the screened macroscopic Coulomb interaction in the long wavelength limit.For 3D systems, since W(q)∝|q| −2 , the LO-TO splitting at Γ point exists only when the Born charge Z * n is nonzero for polar materials.When the dimensionality is reduced to two dimensions, the 2D Coulombscreened interaction can be written as W(q) 2D = 1 2q 0  2D , in which ϵ 2D is the in-plane dielectric function of 2D material, which indicates that, there is no LO-TO splitting at Γ point in 2D materials even for polar materials.
To further investigate the phonon behaviors in these two monolayers, their vibrational modes are calculated and shown in Figure 2 and Figure S1 (Supporting Information), respectively.The small group at Γ point for monolayer MoSi 2 N 4 and WSi 2 N 4 are D 3h, which possesses six irreducible representation (irreps), and the vibrational modes at the Γ point can be decomposed by, where R/I denotes Raman/Infrared activated modes.The vibrational modes for monolayer MoSi 2 N 4 and WSi 2 N 4 are similar due to their identical crystal symmetries.The calculated electronic band structures based on the Perdew-Bruke-Ernzerhof (PBE) method projected by atomic orbitals for monolayer MoSi 2 N 4 and WSi 2 N 4 are shown in Figure 1f,j, which reveals that both monolayers possess indirect bandgaps with the values of 1.79 and 2.11 eV for MoSi 2 N 4 and WSi 2 N 4 , respec-  effects by using the one-shot G 0 W 0 +SOC method, the bandgaps increase to 3.20 and 3.03 eV, correspondingly.The QP energy can be obtained by expanding the electronic self-energy Σ around the Kohn-Sham eigenvalues with the first order, where the normalization factor ,  nk is the single-particle Kohn-Sham (KS) energy of band index n.The self-energy Σ nk can be divided into the exchange item Σ x nk and the energy-dependent correlation item Σ c nk (), and V xc nk is the exchange-correlation potential for |nk〉.In Figure 1j, for monolayer WSi 2 N 4 , when the QP effects are considered, the position of VBM changes from Γ to K point, the value of Σ x nk and Σ c nk () are 1.38 and −17.04 eV at Γ points, 1.29 and −16.71 eV at K points, respectively.This is mainly due to the fact that K point possesses a larger total self-energy correction term compared to Γ point on QP energy, which thus changes monolayer WSi 2 N 4 into a semiconductor with a direct QP bandgap.
To better understand the contribution of atomic orbitals to electronic band structures for monolayer MoSi 2 N 4 and WSi 2 N 4 , the crystal-field splitting and SOC effects at Γ point are considered.According to project atomic orbital around CBM and VBM in Figure 1f,j, we take Mo/W-d orbital into consideration, and the formation of the electronic energy around the Fermi level at Γ for these two monolayers can be summarized into three states, as shown in 1 for valence band at Γ point.At stage (III), when the SOC effect is considered (w SOC), the irreps for the spinor is 2D Γ 7 for double-group D 3h .As shown in Figure 3

Excitonic Behaviors and Excitonic Dynamics
In 2D materials, due to the enhanced quantum confinement effect and thus the reduction of Coulomb screening effects, the contributions from bound electron-hole pairs called excitons  generally play a key role in optical properties.The eigenstate for the S th electron-hole pair can be obtained by solving the Bethe-Salpeter equation (BSE), [20] ( where E QP vk,ck denotes the QP eigenvalues for electronic |vk⟩ and |ck⟩ states, A s vck is the envelop function for the S th exciton, the eigen state of which can be written as the summation over several electron-hole pairs, i.e., |S⟩ = ∑ k,v,c A s vck |ck⟩ ⊗ |vk⟩, and |vk〉/|ck〉 is the constituent electron/hole pair with identical momenta.Ω S is the exciton eigen energy.The contribution to the optical properties from the S th exciton is determined by its oscillator strength, which is defined as , where e is photon polarization vector, v is the velocity operator.
The calculated optical absorptions with and without consideration of the contribution from electron-hole pairs denoted by orange lines and blue lines for monolayer MoSi 2 N 4 and WSi 2 N 4 are shown in Figure 4a,b, respectively, which demonstrates that, in both monolayers, two excitonic absorption peaks can be observed in the QP-bandgap region, labeled by E 1 and E 2 , respectively.The eigenvalues of E 1 and E 2 are 2.45 and 2.61 eV for monolayer MoSi 2 N 4 , which are in good agreement with the experimental results of 2.21 eV, 2.35 eV [21] and theoretical results. [22]The excitonic absorption peaks for monolayer WSi 2 N 4 are 2.29 eV and 2.66 eV, respectively.The calculated eigenvalues and the corresponding oscillator strength of excitons for these two monolayers are shown in Figure 4c,d, in which the oscillator strength is denoted by the radius of bubbles, which reveals that the two absorption peaks in Figure 4a,b are induced by the two bright excitons with large oscillator strengths, also denoted by E 1 and E 2 , respectively.It is worth mentioning that the high oscillation intensity electron-hole pairs (ionization continuum states) on the high-energy side of the bandgap energy also contribute significantly to the absorptions in the high-frequency region of monolayer MoSi 2 N 4 and WSi 2 N 4 , but they are thermally unstable and will dissociate into free charge carriers at finite temperatures.The binding energies of the two bright excitons for these two monolayers, defined as the difference between the exciton eigen energy and the QP bandgap, are 0.75/0.59and 0.74/0.37eV, respectively, both being much larger than thermal fluctuation energy at room temperature of 26 meV, indicating that, E 1 and E 2 excitons for both monolayers are thermally stable at room temperature.In this work, we choose the first bright exciton E 1 for the further study of the dynamics and light-exciton interaction.The electron-hole pairs with dominant contribution In the following, we will discuss the exciton relaxation processes dominated by electron-electron (el-el), exciton-phonon interactions (ex-ph), and radiative recombination.The exciton decays induced by electron-electron interactions including quasielectrons and quasi-holes produced by photoexcitations.Then the el-el lifetime for the S th exciton can be calculated according to  S ee = ( , where the  ck and  vk represent the lifetimes of quasi-electrons and quasi-holes obtained from the self-energy ∑ nk (E QP nk ), which is calculated based on the many-body GW method.The results are shown in Figure 6a,d, which indicates that most of the excitons within the QP bandgaps experience the el-el relaxation processes with a time scale ranging from about 5.0 to 40.0 fs, which are comparable to doped graphene of 40.0-80.0fs. [23]he interactions of excitons and lattice vibrations can be regarded as the superposition of electron-and hole-phonon scattering processes.Based on the first-order perturbation theory, the ex-ph interactions Hamiltonian ΔH ex − ph can be written as, [24] where  mn is the ex-ph coupling matrix element describing the transition probability from the initial S n (Q) exciton with momentum Q to final S m (Q + q) exciton with momentum Q + q of electron-phonon and hole-phonon scattering processes, which can be obtained by Equation (23).ĉ † ∕ d † and ĉ∕ d are the creation and annihilation operators for excitons/phonons, respectively.We take the two ex-ph interaction processes of emission and absorption of phonons into consideration here.To simplify the model we consider that the exciton density F mQ + q is far less than the phonon density N q , and only phonons with zero momentum are involved to calculate.The calculated ex-ph scattering rates for both monolayers are shown in Figure 6b,e, which reveals that optical phonons scattering contributes the most compared to acoustic phonons to the E 1 exciton in monolayer MoSi 2 N 4 and WSi 2 N 4 .The lifetimes of ex-ph coupling are 10 and 3 fs for E 1 exciton in MoSi 2 N 4 and WSi 2 N 4 , respectively, and the values are comparable with the 2D h-BN around 15 fs. [24]inally, the calculated exciton radiative lifetimes  S RA describing the direct recombination between electron and hole according to Equation (27), are shown in Figure 6c,f.The calculated  S RA for E 1 exciton of monolayer MoSi 2 N 4 and WSi 2 N 4 are 25 and 188 ns, respectively, which are much larger than those for black phosphorus of 200 ps, [25] monolayer MoS 2 of around 3 ns with doping. [26]onsidering the feasibility of experimental realization, the recombination lifetime is long enough for the experimental observations of exciton-polaritons.

Phonon and Exciton Polaritons
Generally, the effective Hamiltonian for the interaction of Bosetype quasiparticles (excitons and phonons herein) and photons can be expressed as [27] where a † k (a k ) and b † nk (b nk ) are annihilation (creation) operators of Bose-type quasiparticle phonon or excitons and photon fields with crystal wave vector of k, respectively.g k is the coupling strength of quasiparticles and photons.Following the Bogoliubov-transformation-like method as follows [27] a the interaction Hamiltonian can be diagonalized to the secondquantization form as where E  is the eigenenergies of the hybrid quasiparticle called phonon or exciton polaritons, and  †  (  ) are annihilation (creation) operators of the hybrid polaritons.

2D Phonon Polaritons
As is well known, in bulk polar materials, the long-range polarization field induced by the optical-phonon vibrations influences the phonon vibrations in turn, leading to the LO-TO phonon splitting near the Γ point.On the other hand, the coupling between photons and Bose-type phonons form the phonon polaritons as mentioned above, with the dispersion-relation equation written as, [28] where  LO/TO represents the LO/TO phonon frequency and  is phonon damping rate.The Reststrahlen band is formed between LO and TO phonon frequencies, and there are two dispersion curves for such phonon polaritons, usually called as upper polariton band (UPB) and low polariton band (LPB).When the wave vector k is large enough, the LPB is dominated by the lattice vibration, and the UPB is dominated by photons.When k → 0, the situation is on the opposite.When the dimensionality is reduced to two dimensions, the LO-TO splitting at the Γ point giving rise to the formation of phonon polaritons is absent according to Equation (1) resulting from the 2D dynamical Coulomb screening effects, even for polar monolayers.Strictly, the phonon polaritons are not well defined and therefore absent in any 2D materials.Despite the absence of well-defined phonon polaritons in 2D materials, recent reports demonstrated that, [10,11] the 2D LO phonon modes in the 2D polar materials manifest strongly confined evanescent electromagnetic modes, which is similar to the surface phonon polaritons in 3D systems.Therefore, in this way, the 2D LO phonon modes can be regarded as the equivalence of the surface phonon polaritons, also called 2D phonon polaritons, which was verified previously in 2D hBN [11] and MoS 2 . [9]n 2D materials, by considering the longitudinal phonon distribution to the ionic displacement and then using the conductivity within the random-phase approximation (RPA) method, the lattice optical conductivity in the long-wavelength limit (q → 0) can be written as, [10,11] Here, we consider the monolayer to be sandwiched by a superstrate of permittivity ϵ + and a substrate of permittivity ϵ − , and ϵ env = (ϵ + + ϵ − )/2.Then, the dispersion equation can be reduced to its quasistatic limit without consequential loss of accuracy, i.e., q = 2iϵ 0 ϵ env /().Thus, the dispersion equation for 2D phonon polariton can be written as, [10,11] The calculated Born effective charges Z* for ions in both monolayers are listed in Table 1.The LO phonon group velocities v g for monolayer MoSi 2 N 4 and WSi 2 N 4 , defined from the microscopic parameters by  WSi 2 N 4 hBN [ 11] LO and 18.09, 24.97 THz, respectively, which possess nonzero effective charges, therefore they can contribute to the ionic part of optical conductivities.
The deceleration factor D (= v g /c) for these two infrared-active 2D phonon polaritons in both monolayers are extremely low with 5−7 orders lower than light speed, also lower by 0−2 orders than those in monolayer h-BN, [11] and values of D for lowfrequency LO phonons in these two monolayers are comparable to that in monolayer MoS 2 . [9]The phonon damping rates  for all 2D phonon polaritons are calculated by only considering three-phonon processes by using the Boltzmann transport theory implemented in the ShengBTE package, [29] and similar calculations were previously reported in ref. [30].The calculated  values at room temperature for the two infrared-active 2D phonon polaritons in monolayer MoSi 2 N 4 and WSi 2 N 4 are 10.7, 18.3 cm −1 and 8.3, 16.6 cm −1 , respectively, as listed in Table 2, which are comparable to those in SrTiO 3 and KTaO 3 , but larger than those in LiNbO 3 and h-BN, [11] probably due to the heavier Mo and W atoms.
The electronic conductivities contributed by the infrared-active LO phonon modes can be calculated according to Equation (10), and the calculated real (Re ) and imaginary (Im ) parts of the conductivity  for monolayer MoSi 2 N 4 and WSi 2 N 4 are shown in Figure 7a,b,g,h, respectively, with the mode damping rates listed in Table 2.The linewidth for the peak values of () as shown in Figure 7a,b,g,h is proportional to , and the peak values of  for the LO phonon modes with highest frequencies are larger than those with lower frequencies by one order.
The propagation-quality factors Q, defined as Q = Im /Re , measuring the distance that the 2D phonon polariton travels before decaying in number of wavelengths, are calculated and shown in Figure 7c,d,i,j, respectively, which reveals that the Q shows slower decrease with increasing frequencies for 2D phonon polaritons with larger frequencies and larger damping rates , indicating that the quicker leakage of energy leads to smaller Q.Similar behavior can be observed for all 2D phonon polaritons irrespective of MoSi  KTaO 3 , but slower than those in SrTiO 3 , LiNbO 3 , and h-BN, [11] probably due to their relatively larger damping rate .
Then the confinement factors C for 2D phonon polaritons, defined as C = q/q 0 =  0 / ph with  ph the wavelength for 2D phonon polariton, which is a measure for the compression of light wavelength  trapped within a 2D phonon polariton with respect to light wavelength in free space  0 , are calculated and shown in Figure 7e,f,k,l, respectively.The calculated confinement C for all 2D phonon polaritons in these two monolayers is in the magnitude of 10 3 and 10 5 , i.e.C ≈ O( 103 ) and O(10 5 ), in which the higher value is larger than those in h-BN by two orders, probably due to the smaller deceleration factors D for the lower frequency LO phonon in these two monolayers as listed in Table 2.
In addition, in order to consider the applied monolayer on top of the substrate, the influence of compressive/tensile strains are considered.The calculated band structures and phonon dispersions by applying the strains of −4, −2, +2, +4% along a-axis of the two monolayers are shown in Figures S4 and S5 (Supporting Information).When the strains range from -2% to +2%, the PBE bandgaps are 2.02, 1.52 eV and 2.32, 1.89 eV for monolayer MoSi 2 N 4 and WSi 2 N 4 , respectively, and the phonon frequency gradually redshifts.The calculated phonon properties include the deceleration factor D and phonon damping rate  with the consideration of ±2% strains are shown in Table S1 (Supporting Information), and the optical conductivity , propagation-quality factor Q and confinement factor C for monolayer MoSi 2 N 4 and WSi 2 N 4 are shown in Figure S6 (Supporting Information).Benefiting from the smaller phonon damping rate, the optical conductivity increases by an order of magnitude after applying strain, and the Q value also doubles.Besides, as shown in the Figure S8 and Table S1 (Supporting Information), the properties of the 1Tphase monolayer MoSi 2 N 4 and WSi 2 N 4 are similar to the 2Hphase ones under investigations here, but the 1T-phase monolayers will not be discussed in details here.

Exciton Polaritons
Due to the large binding energies of both E 1 excitons in monolayer MoSi 2 N 4 and WSi 2 N 4 of 0.75 and 0.74 eV, as is shown in Figure 4a,b, which guarantee the thermal stabilities of E 1 excitons against the thermal fluctuation at room temperature, and their long radiative recombination times of 25 and 188 ns, respectively as shown in Figure 6c,f, the E 1 excitons in both monolayers are beneficial for the realization of exciton polaritons and polariton condensation when they are strongly coupled to the cavity photons.Generally, the total exciton energy E X (k) for excitons with finite momentum of k can be given as the summarization of energy of zero-momentum transverse exciton (E X0 ) and the kinetic energy, i.e., E X (k) = E X0 + ℏ 2 k 2 /2M ex , with M ex the mass of exciton given by M ex = m e + m h .The light dispersion in vacuum can be expressed as  0 = ck∕ √  ∞ .For simplification, the influence of the kinetic energy term is neglected for low-frequency cases under investigation.The transverse exciton energies E X0 for E 1 excitons in 2D MoSi 2 N 4 and WSi 2 N 4 are 2.45 and 2.29 eV, respectively, as shown in Figure 4c,d.For enhancement of the excitoncavity photons coupling, assuming the monolayer MoSi 2 N 4 and WSi 2 N 4 are placed in the middle of the cavity with air (n = 1) filled in the space between the distributed Bragg reflectors (DBR), the dispersion for the exciton polaritons can be described from the coupled-oscillator Hamiltonian model as follows, where g is the coupling strength for exciton-photon interaction, also called the Rabi coupling strength, and ℏg S ≈ X ∕ 2 S the Rabi coupling strength/eigen energy/the square modulus of the exciton transition dipole per unit cell for the S th exciton, n 2 = ϵ env , and Ω the area for 2D unit cell.The Rabi splitting for the S th exciton can be obtained as ΔE = 2g.The solution to Equation ( 12) is written as, Subsquently, based on Equation ( 12), the dispersion relation for exciton polaritons in monolayer MoSi 2 N 4 and WSi 2 N 4 with the length L of 100, 200, 300 μm with air filled in the space between the DBRs are calculated, and the results are shown in Figure 8a,d, which demonstrate that both monolayers possess lower polariton band (LPB) and upper polariton band (UPB) as functions of wave vector k and quasiparticle energy, corresponding to ∓ in Equation ( 13), respectively.When the L is 300 μm, the calculated ΔE for monolayer MoSi 2 N 4 and WSi 2 N 4 are 373 and 321 meV, respectively.For comparison, we also calculate their dispersions for exciton polaritons by considering the exciton dipole field, [49] i.e., 2 , with ϵ ∞ the high-frequency constant, and the calculated ΔE are approximately equal to 334 and 319 meV at wave vectors of 29 and 25 MHz for these two monolayers.The comparison of theoretical and experimental Rabi splittings for different materials is shown in Figure 9, which reveals that the Rabi splittings for monolayer MoSi 2 N 4 and WSi 2 N 4 are larger than those of PEA 2 PbI 4 , [16] ZnO, [34][35][36] GaN, [38,40,41] hBN, [38,39] MoS 2 , [38,42,43] WS 2 , [17,38,44] MoSe 2 , [38,45,46] WSe 2 , [38,47,48] and Phosporene, [31] but comparable to bulk (C 6 H 5 C 2 H 4 NH 3 ) 2 PbI 4 (PEPI) [37] and Ag, [32,33] manifesting their superiority in terms of interaction between light and excitons.Furthermore, when the length of cavity decreases from 300 to 200 μm, the Rabi splitting is changed to be 457 and 393 meV, and further decreases to 100 μm, the Rabi splitting is increased to 646 and 556 meV, respectively, indicating stronger light-matter interactions for smaller cavity lengths.
The calculated properties of exciton and exciton polariton with the consideration of ±2% strains are shown in Figure S7 (Supporting Information).The GW bandgap of −2, +2% strain for monolayer MoSi 2 N 4 are 3.7 and 3.1 eV, and the values are 3.6 and 3.2 eV for WSi 2 N 4 , which both increase, compared with those without strains.The Rabi splitting are 327, 185 and 255, 226 meV for the two monolayers, respectively, which are smaller than those without strains.
To manifest the respective contribution from photons and excitons to the formation of exciton polaritons, the annihilation operator for exciton polariton, i.e.,   in Equation (8), can be written as a linear combination of annihilation operators for photon and exciton, [50]  LP (k  [16] Phosporene, [31] Ag, [32,33] ZnO, [34][35][36] PEPI, [37] hBN, [38,39] GaN, [38,40,41] MoS 2 , [38,42,43] WS 2 , [17,38,44] MoSe 2 , [38,45,46] WSe 2 . [38,47,48]ere the prefactors  C (k) and  X (k) are known as Hopfield coefficients for cavity photons and excitons, respectively.The squares of the Hopfield coefficients in each UPB and LPB give the fraction of exciton and photon in exciton polariton, and the Bose commutation relation yields  C (k) 2 +  X (k) 2 = 1.When the strong coupling occurs between exciton and photon, the square of the Hopfield coefficient of the exciton and photon is equal to 0.5, which means that the exciton polariton is formed by half photon and half exciton, also corresponding to the location where the Rabi splitting occurs, i.e., at wave vectors of 29 and 25 MHz as shown in Figure 8b,e.With the increase of the wave vector, the Hopfield coefficient of the exciton  X (k) increases, but that for cavity photon  C (k) decreases, which means that, the LPB-branch polaritons become more exciton-like, whereas the UPB-branch polaritons become more photon-like.When the wave vector decreases, the situation is the opposite.The polariton effective mass of LPB/UPB can be written as the weighted harmonic mean of the masses of its constituent exciton and photon as follows, [51] 1 where m C is the effective mass for the DBR cavity photons with the value ≈10 Strictly speaking, the lifetime of polaritons  LP, UP is determined by the Rabi coupling strength g, lifetimes of cavity photons due to the imperfect mirrors  C , and lifetimes of excitons  X mentioned above, and in microcavities with ) is the decay rate of cavity photons and excitons, respectively, the lifetime of quantized exciton polariton can be expressed as a linear combination of cavity photon lifetime and exciton radiative lifetime, [51] i.e., 1∕ As shown in Figure 6c,f, the calculated lifetimes of the first bright excitons  X for monolayer MoSi 2 N 4 and WSi 2 N 4 are 25 and 188 ns, respectively.Generally, the lifetimes of the cavity photon  C are 1-10 ps.Therefore, X and at the location where the Rabi splitting occurs,  UP, LP ≈ 2 C , with the time scale of ps.

Conclusion
In summary, based on the first principle and many-body perturbation theory, we theoretically analyzed 2D phonon vibration mode and exciton dynamic processes of monolayer MoSi 2 N 4 and WSi 2 N 4 .We have demonstrated the crystal-field splitting and SOC effects at Γ point.Then we examine the properties of two infrared active phonon polaritons.The propagation-quality factors Q tendency for monolayer MoSi 2 N 4 and WSi 2 N 4 are comparable to those in KTaO 3 and the confinement factors C are larger than those in h-BN by zero to two orders.Besides, we extensively demonstrate the 2D exciton polaritons of monolayer MoSi 2 N 4 and WSi 2 N 4 .The large binding energy of the first bright exciton are 750 and 740 meV, and the calculated long exciton radiative lifetime is 25 and 188 ns, respectively, which is beneficial for the implementation of cavity exciton polaritons.We have calculated the strong Rabi splitting of the first bright exciton, the values are 373 and 321 meV, larger than many conventional polariton materials.

Experimental Section
The calculations were based on the first principles implemented using the Vienna Ab Initio simulation package (VASP) and PHONOPY.The 2D implementation of the nonanalytical-term correction was performed to get the correct dispersion of the LO phonon by Quantum ESPRESSO.For the calculations on electronic structures and structural optimization, the Generalized gradient approximation (GGA) in the PBE parametrization was used to describe the exchange-correlation potentials.A 21 × 21 × 1 Gamma-centred reciprocal-space grid was used for the K-points sampling.A convergent 24 × 24 × 1 k-mesh was used to calculate QP states for monolayer MoSi 2 N 4 and WSi 2 N 4 , based on the many-body interaction G 0 W 0 method implemented in the YAMBO software package.The kinetic cutoff energy of the plane waves was set to be 90 Ry.
Furthermore, many-body perturbation theory was used to perform the electron correlation calculations.The Hamiltonian considering interacting electrons system can be expressed as the following Dyson equation, [52] [ QP nk (16)   where V H is the Hartree term, V DFT xc is the DFT exchange-correlation potential, E QP nk and  QP nk represent QP eigen-energies and eigen-wavefunctions, respectively.And the self-energy operator Σ describes electron-electron interactions, which can be approximately obtained by Σ ≈ iG 0 W 0 , [52,53] where the Green's function G 0 is constructed by KS orbitals, and W 0 is the frequency-dependent dynamically screened interaction defined as W 0 = [1 − VP 0 ] −1 V, where P 0 and V represent polarizability and bare Coulomb interaction, respectively.The imaginary part of the electron self-energy Σ nk represents the lifetime due to the quasi-particle effect, and the QP lifetime for |nk > electron is calculated according to Due to the weak Coulomb screening effect in 2D systems, the electronhole (exciton) interactions were enhanced significantly.The contribution from the electron-hole pair of monolayer MoSi 2 N 4 and WSi 2 N 4 should be involved in the calculation of optical properties.We further solve the Bethe-Salpeter equation (BSE) for the excitonic state, (19)   here K eh includes the kernel of electron-hole interaction, Ω S and A S vck represent the eigen-energies and momentum-space envelop function for the S th exciton.The real space wave function for the Sth exciton can be expressed as A S vck  k,c (r e ) * k,v (r h ) Then, the imaginary part of the dielectric function ϵ 2 can be calculated by summarizing over certain number of excitons, [54]  2 () where e ⋅ ⟨0|v|S⟩ = Σ vck A S vck ⟨vk|v|ck⟩, with v representing the velocity operator of the incident photons.In this work, six lowest QP conduction bands and six QP valence bands were used.
Similar to equation, the lifetime limited by the electron-electron interactions for the S th exciton can be written as, [23] 1 Furthermore, the relaxation process via phonons for excitons involves electron-phonon interactions, and the electron-phonon matrix element for the transition between the electron |nk〉 and |mk + q〉 via the phonon |qv〉 can be calculated by, And the exciton phonon matrix elements  mnv (Q, q) can be expressed as, [24]  mnv (Q, q) = ∑ where A m vk,c ′ (k+Q+q) and A n vk,c(k+Q) are the exciton envelop functions.Using Matsubara's method, the imaginary part of exciton-phonon self-energy can be calculated, where N vq and F mQ+q are phonon and exciton occupations and N q and N k are the numbers of q points and k points.Using free exciton approximation, the self-energy is reduced as, [55] Ξ The excitons may experience the direct recombination decay which can be estimated as follows, [56]  S (Q where  2 S is the velocity dipole matrix element expressed by, where m e is the electron mass, 〈G|p ‖ |Ψ S (0)〉 is the momentum dipole matrix element in the exciton basis, and N k is the number of k vector used in the calculation.

Figure 1 .
Figure 1.a) Crystal structure of MoSi 2 N 4 and WSi 2 N 4 monolayers.b) Top and c) side views.The phonon dispersions and phonon density of states of d,e) MoSi 2 N 4 and h,i) WSi 2 N 4 .The project band structures and PDOS of f,g) MoSi 2 N 4 and j,k) WSi 2 N 4 .The gray lines are G 0 W 0 band structure.

Figure 2 .
Figure 2. Vibration analysis for phonon modes at the Γ point of monolayer MoSi 2 N 4 .

Figure 3 .
At the stage (I), the chemical bonding separates the five-fold degenerate Mo/W-d atomic orbitals into the low-energy |d + 〉 bonding states and high-energy |d − 〉 antibonding states, and the +/− also means even/odd parity regarding the in-plane mirror symmetric plane.At the stage (II), without the consideration of SOC effect (w/o SOC), the crystal-field effect restricted by the D 3h point group, splits the fivefold degenerate low-energy d + orbital into twofold degenerate (d + xz , d + yz ) transforming with E″, and twofold degenerate (d + xy , d + x 2 −y 2 ) transforming with E′ for conduction band, and singly degenerate d + z 2 energy levels transforming with A ′

ⅠFigure 3 .
Figure 3. Schematic diagram of the evolution of the atomic Mo/W-d and N-p orbitals of monolayer MoSi 2 N 4 and WSi 2 N 4 into the valence and conduction bands, the red dotted line represents the Fermi energy level.Ψ(j, m j ) means the basis functions that transform like an eigenstate of the angular momentum operator J of total angular momentum J and z component m j .

Figure 4 .
Figure 4. a,b) Energy-dependent optical absorption of monolayers MoSi 2 N 4 and WSi 2 N 4 calculated by G 0 W 0 +BSE (orange line) and G 0 W 0 +RPA (blue line).c,d) Energy-dependent excitons with different binding energies for monolayers MoSi 2 N 4 and WSi 2 N 4 with the circle radius representing oscillator strengths for respective excitons.e,f) Exciton energy levels for monolayers MoSi 2 N 4 and WSi 2 N 4 .

Figure 5 .
Figure 5. Exciton wavefuctions in real space of E 1 exciton for monolayer a) MoSi 2 N 4 , b) WSi 2 N 4 , with the hole at center of supercells.
2 N 4 or WSi 2 N 4 .The Q tendency for monolayer MoSi 2 N 4 and WSi 2 N 4 are comparable to those in

Table 1 .
Optimized lattice constants a, eigenvalues of excitons E x , highfrequency dielectric constant ϵ ∞ and Born effective charges Z*.

Table 2 ,
are calculated and v g for these two 2D materials are listed in in which c is the light speed in vacuum.As mentioned above, there exist two infrared-active LO phonon modes in monolayer MoSi 2 N 4 and WSi 2 N 4 with the frequencies of 18.21, 25.72 THz,

Table 2 .
The LO phonon frequencies, the deceleration factor D and phonon damping rate  for two infrared-active LO/TO phonon modes in monolayer MoSi 2 N 4 and WSi 2 N 4 , one infrared-active LO/TO phonon modes in monolayer hBN.