Cavity exciton-polaritons in two-dimensional semiconductors from first principles

Two-dimensional (2D) semiconducting microcavity, where exciton-polaritons can be formed, constitues a promising setup for exploring and manipulating various regimes of light-matter interaction. Here, the coupling between 2D excitons and metallic cavity photons is studied by using first-principles propagator technique. The strength of exciton-photon coupling is characterised by its Rabi splitting to two exciton-polaritons, which can be tuned by cavity thickness. Maximum splitting of 128 meV is achieved in phosporene cavity, while remarkable value of about 440 meV is predicted in monolayer hBN device. The obtained Rabi splittings in WS$_2$ microcavity are in excellent agreement with the recent experiments. Present methodology can aid in predicting and proposing potential setups for trapping robust 2D exciton-polariton condensates.

Two-dimensional (2D) semiconducting microcavity, where exciton-polaritons can be formed, constitues a promising setup for exploring and manipulating various regimes of light-matter interaction. Here, the coupling between 2D excitons and metallic cavity photons is studied by using first-principles propagator technique. The strength of exciton-photon coupling is characterised by its Rabi splitting to two exciton-polaritons, which can be tuned by cavity thickness. Maximum splitting of 128 meV is achieved in phosporene cavity, while remarkable value of about 440 meV is predicted in monolayer hBN device. The obtained Rabi splittings in WS2 microcavity are in excellent agreement with the recent experiments. Present methodology can aid in predicting and proposing potential setups for trapping robust 2D exciton-polariton condensates.
Despite this enourmous interest in exciton-polaritons and seminconductor microcavity devices, a complementary microscopic theories that are able to scrutinize the cavity photon-exciton coupling on the quantitative and predictive level are still rare. In addition, the majority of the microscopic descriptions are based on simple model Hamiltonians describing exciton-photon interactions in microcavity [34][35][36][37][38][39]. Recently, a more rigorous ab initio theoretical description of exciton-polaritons in TMD microcavity was provided in the framework of the quantum-electrodynamical Bethe-Salpeter equation [40], where the excitons are calculated from first-principles Bethe-Salpeter equation, and the electromagnetic field is described by quantized photons. However, the coupling between excitons and photons is left to be arbitrary. This study showed how excitonic optical activity and energetic ordering can be controlled via cavity size, light-matter coupling strength, and dielectric environment.
Here, we present a fully quantiative theory of 2D exciton-polaritons embedded in plasmonic microcavity that is able to analyze and predict light-matter coupling strengths for various cavity settings. We study cavity exciton-polaritons in three prototypical two-dimensional single-layer semiconductors, i.e., single-layer black phosphorus or phosphorene (P 4 ), WS 2 , and hexagonal boron nitride hBN. The results show a clear Rabi splitting between 2D exciton and cavity photon modes as well as high degree of tunability of Rabi (light-matter) coupling Ω as a function of microcavity thickness. In the case of WS 2 , for the usual experimental setup where cavity size is around d ∼ 1 µm we obtain splittings of Ω ∼ 42 meV and Ω ∼ 64 meV for principal and second cavity modes, in line with the experiments [25,27]. Also, in all cases larger coupling strengths Ω are found for larger photon confinements when d < 1 µm as well as for higher cavity modes n. Interestingly, the ultraviolet (UV) exciton in hBN shows a very strong exciton-cavity photon coupling of about ∼ 440 meV and a possiblity of Bose-Einstein condensation.
In this work both excitons and photons are described by bosonic propagators σ and Γ, repectively, which are  derived from first principles. The 2D crystal optical conductivity σ is calculated using ab initio RPA+ladder method [41] , and propagator of cavity photons Γ is derived by solving the Maxwell's equations for planar cavity decribed by local dielectric function (see Supplemental Material [42]). The exciton-photon coupling is achieved by dressing the cavity-photon propagator Γ with excitons at the RPA level. Thus obtained results are therefore directly comparable with the experiments. As illustrated in Fig. 1(a) the microcavity device consists of substrate, tip and dielectric media in between, described by local dielectric functions + , − and 0 , respctively. The 2D semiconducting crystal defined by optical conductivity σ µ (ω) is immersed in a dielectric media at a height z 0 relative to the substrate. The substrate occupy region z < 0, tip occupy region z > d, and dielectric media occupi region 0 < z < d. In such semiconducting microcavity setup the coupling between the exciton and cavity photon is expected to result in the splitting of exciton-polariton to the lower and upper polariton branches (LPB and UPB), which we shall refer as ω − n and ω + n , respectively [see Fig. 1 The quantity from which we extract the information about the electromagnatic modes in microcavity setup is electrical field propagator E µν which, by definition [43], propagates the electrical field produced by point oscillating dipole p 0 e −iωt , i.e. E(ω) = E(ω)p 0 . Assuming that the 2D crystal, substrate and tip satisfy planar symmetry (in x − y plane) the propagator E in z = z 0 plane satisfies matrix equation as illustrated by Feynmans diagrams in Fig. 2(a) (see also Sec. S1.A in Ref. [42]). Here Γ = Γ 0 + Γ sc represents the propagator of electrical field, in absence of 2D crystal, ie. when σ = 0. The propagator Γ 0 represents the "free" electrical field and the propagator of scattered electrical field Γ sc results in multiple reflections at the microcavity interfaces, as ilustrated in Fig. 2(b). In order to simplify the interpretation of the results we suppose that the dielectric media is vacuum ( 0 = 1), and we suppose that tip and substrate are made of the same material ( − = + = ). In order to support well-defined cavity modes, these materials should be highly reflective in the exciton frequency region ω ≈ ω ex , which is satisfied if ω ex < ω p , where ω p is the bulk plasmon frequency. For the P 4 and WS 2 monolayers where exciton energies are ω ex < 3.0 eV, we chose that the substrate and tip are made of silver (ω p ≈ 3.6 eV). One the other hand, for the single-layer hBN where exciton energy is ω ex = 5.67 eV we chose aluminium (ω p ≈ 15 eV). Both silver and aluminium macroscopic dielectric functions (ω) are determined as well from the first principles (see Ref. [42]).

Figures 3(a), 3(b), and 3(c)
show the modifications of the n = 1 cavity mode intensity after the single-layer P 4 is inserted in the middle z 0 =d/2 (n = 1 antinodal plane) of the silver cavity, where the cavity sizes are d = 375 nm, d = 400 nm and d = 425 nm, respectively. White and turquoise dotted lines denote the P 4 exciton and unperturbed cavity mode n = 1, respectively. For d = 375 nm, just before the n = 1 cavity mode crosses the exciton, a significant part of the n = 1 mode spectral weight is transferred below the exciton energy. By increasing the cavity size, i.e. for d = 400 nm and d = 425 nm, the exciton crosses the n = 1 mode, which results in the intensity weakening and band-gap oppening in the intersection area. This behaviour enables creation of exciton-polariton condensate, as experimentaly verified in Refs. [6,[13][14][15]29]. By changing the cavity thickness, the exciton can interact also with the higher cavity modes. Figure 3(d) shows the modifications of n = 2 mode intensity, where the cavity thickness is d = 850 nm and P 4 is choosen to be located at z 0 = d/4nm (n = 2 antinodal plane). The exciton significantly weakens the intensity of n = 2 mode in the intersection area, however, here the avoided crossing behaviour is not clearly noticeable in comparison with the exciton coupled to the 1st cavity mode.
The dispersion relation of exciton-polaritons ω − n and ω + n (hybridised cavity photon-exciton modes), as the one shown in Fig. 4(a), can be precisely determined by following the splitted maxima in induced current j µ = σ scr µ E µ driven by external (bare) field E µ e −iωt , where the screened optical conductivity is σ scr µ = [1 − Γσ] −1 µ σ µ . The inset of Figure 4(b) shows the Reσ scr x before (brown dashed) and after (solid magenta) the P 4 is inserted in the middle of the cavity of thickness d = 400 nm. The spliting of exciton ω ex to exciton-polaritons ω − 1 and ω + 1 can be clearly seen. The exciton-photon binding strength can be determined from the Rabi splitting defined as difference Ω n = ω + n − ω − n for wave vector Q and for which the bare cavity modes n = 1, 2, 3, ... crosses the exciton ω ex . Figure 4(a) shows the dispersion relations of plasmon-polaritons ω − 1 and ω + 1 obtained by following the splitted maxima in Re σ scr x for different wave vectors Q x , d = 400 nm and z 0 = d/2. The clear anticrossing behaviour and Rabi splitting of Ω 1 = 123 meV indicates strong interaction between exciton and n = 1 cavity photon. Red circles, yellow squares and green triangles in Fig. 4(b) show the Rabi splittings Ω n for n = 1, n = 2 and n = 3, respectively, versus cavity thickness d. The maximum Rabi splittings of nth mode Ω max n are achieved when z 0 = d/2 and for d choosen so that nth mode just starts to cross the exciton energy ω ex . All three modes show strong coupling with exciton that results in the as n increases confirms a confinement hypothesis; as n increases the cavity photon modes crosses the exciton for larger d, and photon becomes less confined while the coupling is reduced. Thus, the coupling will be stronger as the thickness d at which the crossing between nth mode and exciton occurs is getting smaller.
The above criterion is met by excitons with higher excitation energy, such as for instance the UV exciton in the hBN single layer. Since in the same UV frequency region the cavity should be highly reflective (i.e., ω p > ω ex ), the appropraite cavity for hBN layer can be made of aluminium with ω p ≈ 15 eV. Figure 5 The 2D exciton-polaritons are experimentally studied mostly in various TMDs where the mesured Rabi splittings of Ω = 46 meV, Ω = 26 meV, and Ω = 20 meV are found in MoS 2 [25], WSe 2 [28], and MoSe 2 [26]. For WS 2 the experimentally measured splittings are around 20 − 70 meV for d > 1 µm, depending on the precise cavity size [27]. In Fig. 5(b) we show the modification of the silver n = 1 cavity mode intensity when the WS 2 monolayer is inserted in the middle of microcavity of thicknesse d = 260 nm. The unperturbed n = 1 mode as well as the A and B excitons of bare WS 2 are also denoted by dotted lines. Both excitons significantly per-turbe the n = 1 mode providing the Rabi splitings of Ω A 1 = 117 meV and Ω B 1 = 103 meV. For comparison, Fig. 5(c) shows the modification of n = 1 mode intensity when P 4 is inserted in silver microcavity for the same conditions as in WS 2 microcavity presented in Fig. 5(b); n = 1 minimum is 100 meV below the exciton. The cavity thickness is d = 415 nm and z 0 = d/2. Interestingly, the achieved Rabi splittig is here also Ω 1 = 117 meV, even though according to confinement hypotesis, the A exciton, which is confined in smaller cavity, is expected to split more. However, the A exciton in WS 2 has smaller oscillatory strength than P 4 exciton [cf. Figs. S4(a) and S5(b) in Ref. [42]] so that the binding is weaker and the two effects cancel.
Finally, in Fig. 5(d) we compare the maximum splittings of exciton-polaritons Ω max for the three semiconducting microcavities, summarizing the different regimes of exciton-cavity photon coupling strengths in these materials. For WS 2 microcavity we additionaly present the results for the experimentally measured value of cavity size, i.e., d = 930 nm [27], when exciton interacts with n = 2 cavity mode. The corresponding value of Ω A 2 = 64 meV shows an excellent agreement with the experiment [27]. For the same cavity thickness, splitting of the n = 1 cavity mode and the WS 2 A exciton is Ω A 1 = 42 meV.
In summary, we have studied the intercation strengths between cavity photons and excitons in various 2D semiconducting crystals by means of rigorous ab initio methodology. It is shown that insertion of 2D crystals into a metallic microcavity significantly modifies the photon dispersion. For instance, the band gap opening and Rabi splitting as high as Ω = 440 meV was obtained for hBN cavity device. This opens a possibility of experimental realization of the robust 2D excitonpolariton condensate. Moreover, the exciton-photon interaction strongly depends on photon confinement, which was shown to be adjustable by the cavity thickness d. The results of exciton-polariton splitting in WS 2 cavity device show a good agreement with recent experiments and suggest higher photon confinements with decreasing cavity size at which stronger photon-matter coupling should be achieved. In order reach this stronger binding we suggest an experimental setup consisting of tunable submicrometer cavity (such as AFM-tip and substrate) tuned so that the principal photon cavity mode coincide with the exciton energy, e.g. as in Fig.4(b).
The authors acknowledge financial support from European Regional Development Fund for the "QuantiXLie