Excitons in hexagonal boron nitride single-layer: a new platform for polaritonics in the ultraviolet

Excitons in hexagonal boron nitride single-layer: a new platform for polaritonics in the ultraviolet F. Ferreira and A. J. Chaves Centro de Física and Departamento de Física, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal N. M. R. Peres and R. M. Ribeiro Centro de Física and Departamento de Física and QuantaLab, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal and International Iberian Nanotechnology Laboratory (INL), Av. Mestre José Veiga, 4715-330 Braga, Portugal (Dated: July 18, 2018) The electronic and optical properties of 2D hexagonal boron nitride are studied using first principle calculations. GW and BSE methods are employed in order to predict with better accuracy the excited and excitonic properties of this material. We determine the values of the band gap, optical gap, excitonic binding energies and analyse the excitonic wave functions. We also calculate the exciton energies following an equation of motion formalism and the Elliot formula, and find a very good agreement with the GW+BSE method. The optical properties are studied for both the TM and TE modes, showing that 2D hBN is a good candidate to polaritonics in the UV range. In particular it is shown that a single layer of h-BN can act as an almost perfect mirror for ultraviolet electromagnetic radiation. I. INTRODUCTION Two dimensional hexagonal boron nitride (hBN), also called by some white graphene, is an electrical insulator in which the boron (B) and nitrogen (N) atoms are arranged in a honeycomb lattice and are bounded by strong covalent bonds. Like graphene, hBN has good mechanical properties1and high thermal conductivity.2 Specially interesting is the possibility of using hBN as a buffer layer in van der Waals heterostructures, namelly ones comprised by layers of h-BN/graphene.3Hexagonal boron nitride layer can serve as a dielectric or a substrate material for graphene in order to improve its mobility4 and open a gap5. It can also be used to improve the thermoelectric performance of graphene.6 Yet, its electronic properties differ significantly from graphene. Graphene πand π∗electronic bands have a linear dispersion at the K point, whereas in hBN there is a lift of the degeneracy at the same point and a wide band gap greater than 7 eV is formed, at least within an independent electron picture. That would, in principle, make it ideal for optoelectronic devices in the deep ultraviolet region7,8. As we will see, however, excitonic effects play an important role in this material: excitonic peaks are created at the near UV, and this is a much more useful electromagnetic spectral range, when compared to the deep UV. The optical properties of monolayer hBN at the UV range are characterized by the exciton with a corresponding optical band gap calculated in the range 5.30–6.30 eV (see Sec. II). The presence of the exciton in this range can be used to excite exciton-polaritons, that share some properties with surface plasmon-polaritons9,10. Therefore, the UV optical properties of hBN can be used as an alternative to the emerging field of UV plasmonics.11–20 The plasmonics in UV range also attracts interest in biological tissue21 as consequence of the resonances in nucleotide bases and aromatic amino acids. Plasmonics in this ranges relies in poor metals13,15,16,22 and Rhodium17,18,20. Because of the difficulty of its synthesis, few experimental works have been done for hBN single layer. Also, to study and probe its electronic and optical properties it is necessary to work in UV range. To our best knowledge only one experimental work23 has been produced that studies the electronic properties of 2D hBN. Those authors observed the band structure of BN monolayer on Ni(111) surface by using angle-resolved ultraviolet-photoelectron spectroscopy and angle-resolved secondary-electron-emission spectroscopy. Because the bond between the interface of h-BN and Ni(111) is weak, the band-structure observed can be regarded as that of the monolayer h-BN. The band gap was determined to be ∼7 eV and after a comparison with theoretical works, the authors conclude that the band gap is estimated to be within the range of 4.6 to 7.0 eV, too wide when compared with numerical results. These theoretical works were based on first principles calculations using Density Functional Theory (DFT). It is well known that DFT does not predict with good accuracy the electronic and optical properties of semiconductors and insulators. Accurate values require a theory that include many-body effects like the GW approximation.24,25 To obtain optical properties, a theory that includes the excitonic properties is also needed. Usually, the Bethe-Salpeter equation (BSE)26,27 is used. There are several works in the literature that used the GW approximation28–30 and GW+BSE31–33 on 2D hBN. The results from these works vary significantly, as can be seen in Table I. There is no agreement even on whether the gap is direct or indirect. Convergence can be an issue in GW and BSE calculations as can be seen arXiv:1807.06413v1 [cond-mat.mes-hall] 17 Jul 2018 2 in References 34 and 35. It is likely that the works summarized in Table Iuse different criteria for convergence and that may explain the differences. A small number of bands used in the calculation28,29 or not using a truncation to avoid interaction with periodic images30 may also explain some differences. Sometimes there is some ambiguity between the value stated for the gap and the one that can be obtained from the absorption spectrum presented.32 More difficult to explain are the values obtained in Ref. 33. They differ significantly from our work and others, although they seem to have converged the calculations carefully. One explanation may be that they fixed the lattice constant at the experimental value, instead of relaxing the unit cell. The experimental lattice constant may not match the value that actually optimizes the system and can influence the values of the gaps in the electronic band-structure. An effectiveenergy technique36 was adopted in Ref. 31. That technique allowed the calculation of the screened Coulomb interaction Wto be converged with only 90 bands and 60 bands for the self-energy Σcalculation. The use of such technique certainly will produce some differences in the final results. In this work we clarify whether the gap in hBN is direct or indirect, as well as their values and the exciton energies. We also calculate the excitonic spectra using an equation of motion formalism and the Elliot formula, fitting it with the GW+BSE calculations thus obtaining a validation of the method. In Section II we describe the details of the G0W0calculations and results. In Section III we show the results of the BSE calculations. Both G0W0and BSE calculations were performed with the software package BerkeleyGW.37–39 Section IV presents the equation of motion formalism and the results for the excitonic properties of monolayer hBN. In Section Vwe study the properties of exciton-polaritons of hBN and we show that a monolayer of hBN can be used as a UV mirror. We finally draw the conclusions in Section VI. Table I. Several band gaps calculated in this work and by other authors using GW0,G0W0and BSE, in eV. K→K indicates the direct gap at the K point, K→Γindicates the indirect gap from the K to the Γpoint, O. Gap is the optical gap and EBE the excitonic binding energy. Reference Calculation K→K K→ΓO. Gap EBE This work G0W0+ BSE 7.77 7.32 5.58 2.19 Ref. 32 GW0+ BSE 7.80 - 6.30 2.10 or 1.50 Ref. 31 G0W0+ BSE 7.36 - 5.30 2.06 Ref. 33 G0W0+ BSE 7.25 - - 1.90 Ref. 30 G0W07.40 - - Ref. 29 GW06.86 - - Ref. 28 G0W06.00 - - Figure 1. (Color online) Electronic band structure (left) and electronic density of states of h-BN (right) for both DFT and GW calculations. II. G0W0RESULTS G0W0calculations were done on top of DFT calculations with a scalar-relativistic norm-conserving pseudopotential. The software package Quantum ESPRESSO40 was used for the DFT calculations. The details of the DFT calculations are summarized in Table II. For G0W0calculations, a truncation technique is needed due to the non-local nature of this theory. We found that for DFT calculations a grid of 6×6×1 k-points is enough to reach convergence. For the GW calculations, a grid of 16 ×16 ×1k-points and a cut off energy of 22.6 Ry and 1100 bands were needed for the dielectric matrix calculations. For the Σself-energy calculation we used a cut off energy of 22.6 Ry and 1000 bands. The results obtained for the electronic band gap are summarized in Table III. They show that a monolayer of hBN is a wide band-gap indirect-gap material. Fig. 1presents the electronic band structure and electronic density of states for both DFT and GW calculations. As mentioned in the Introduction, the only experimental work we are aware off is the one from Ref. 23, which in fact estimates the band gap based on theoretical works that used mean field calculations to predict the electronic properties of bulk h-BN. Mean field theories such as DFT underestimate the band gap value of semiconductors and insulator materials. They obtain a wide range of possible values, from 4.6 to 7.0 eV. We believe that a value closer to 7.0 eV is more reliable, since the gap value for the Table II. Details of DFT calculations. Exchange-correlation functional GGA-PBE41 Plane-wave cut-off 70 Ry K-point sampling (Monkhorst-Pack)42 6×6×1 Interlayer distance 15.8 Å Lattice constant 2.5 Å 3 Table III. G0W0gap values for the transitions K→Γ, K→K and Γ→Γ, and optical gap and exciton binding energy (EBE) obtained from BSE in this work. The results show that hBN is an indirect-gap insulator. Transition K→ΓK→KΓ→ΓOptical EBE Energy [eV] 7.32 7.77 9.07 5.58 2.19 bulk materials are lower when compared to the monolayer counterpart. And is actually closer to the ones obtained by works referred in Table I. Still, more experimental work is needed. Ref. 23 also calculated the width of the valence bands, and they found no good agreement with theoretical works of the time. Table IV shows the width of the valence bands as calculated with DFT, GW and the experimental determination of Ref. 23. The π-band is the one that has its highest energy at the K-point, while the σ1and σ2are the bands that have the highest energy at the Γ point. Table IV shows that DFT results differ from the experimental ones by values greater than 0.5 eV in all cases. On the other hand, G0W0results differ from the experimental results by values equal or smaller than 0.1 eV. We also calculated the effective masses of the highest valence band and lowest conduction band using both G0W0and DFT (Table V). We found no differences between G0W0and DFT, except for the effective mass at K→Γon the first conduction band (DFT value greater by 0.08me). Thus we conclude that DFT calculations are reliable to obtain the values of the effective masses in this material. Ref. 28 also calculated the effective mass at the Γ point for the conduction band, and obtained a value of (0.95 ±0.05)mewith only slight variations for different planar directions. In our work we obtained differences of 0.3mebetween different directions in reciprocal space at the Γpoint. III. BSE RESULTS After determining the conduction and valence band states, the electron-hole pair states are determined using the Bethe-Salpeter (BSE) equation. The imaginary Table IV. Width of the valence bands. πband is the top valence band at K point. σ1and σ2are the bands that have the highest energy at the Γpoint. The difference is in the width of the bands which is greater for σ2. Width of bands [eV] πband σ1band σ2band This work (DFT) 5.20 5.78 7.49 This work (G0W0) 5.90 6.42 8.24 Experimental23 5.80 6.50 8.20 Table V. Effective masses (in electron mass (me) units) for hBN calculated using G0W0. The arrow indicate the direction in which the effective mass is calculated. Effective mass m∗/me Symmetry points K→ΓΓ→KΓ→MM→Γ Valence band 0.63 0.82 1.09 0.46 Conduction band 0.83 0.95 1.27 0.35 part of the dielectric function 2(ω)is then39 2(ω) = 8π2e2 ω2X S |e· h0|v|Si|2δ(ω−ΩS)(1) where ΩSis the energy for an excitonic state S,h0|v|Si is the velocity matrix element, and eis the direction of the polarization of incident light with energy ω.eis the electron charge. If we do not consider excitonic effects, the expression becomes a transition between single particle states39 2(ω) = 8π2e2 ω2X vck |e· hvk|v|cki|2δ(ω−(Eck−Evk)) (2) which is a random phase approximation (RPA). The labels v(c)denotes valence (conduction) band states, and kdenotes the single particle momentum (only vertical transitions are considered). Fig. 2shows the imaginary part of the dielectric function calculated by BSE, done on top of a G0W0calculation with a grid of 16 ×16 ×1k-points. The convergence of the G0W0band structure with a particular grid of k-points does not imply that BSE will be converged with the same grid. An interpolation with a fine grid of 120 ×120 ×1k-points was needed to achieve convergence. Fig. 2also shows the imaginary part of dielectric function without excitonic effects. The first peak has an energy of 5.58 eV and the second peak has an energy of 6.48 eV. In Table III we summarize the gap values of the band structure, the optical gap and the excitonic binding energy. Fig. 3shows the real part of the dielectric function calculated with and without excitonic effects. We also calculated the eigenvalues of the two particle states. Figure 4shows the energies of the 8 lowest energy excitonic states. From now we label each state by the corresponding energy in an ascending order. The pairs of states (1,2), (3,4), and (7,8) are degenerate. States 1 and 2 are the degenerated ground state. We plot the probability density |φ(re,rh)|2obtained from the BSE for these eight excitonic states in Fig. 5. These plots show the probability to find an electron at position reif the hole is located at rh. We set the hole localized slightly above the nitrogen atom. The results were calculated using a coarse grid of 12 ×12 ×1k-points and a BSE interpolation of 72×72×1k-points. It can be noticed the complementarity of the degenerate states. For instance, 4 Figure 2. (Color online) Imaginary part of dielectric function of 2D h-BN. The blue (red) lines represent the BSE (RPA) imaginary part of dielectric function. Figure 3. (Color online) Real part of the dielectric function of 2D h-BN. The blue (red) lines represent the BSE (RPA) real part of dielectric function. if one adds the probability density of states 3 and 4, the symmetry of the lattice is recovered. And the same can be seen for the other degenerate states. The work of Ref. 33 has also studied the excitonic states. Their results are in good agreement with the ones obtained from this work. IV. BSE IN THE EQUATION OF MOTION FORMALISM AND THE ELLIOT FORMULA In this section we will follow the approach of the equation of motion derived in Ref. 43 and detailed in the Appendix A. The formalism is grounded on the calculation of the expected value of the polarization operator ˆ P(t)after we introduce an external electric field of intensity E0and frequency ωthat couples with the electron gas in the 2D material. The optical conductivity and other properties can be obtained from the macroscopic relations. The starting point of our model is an effective Dirac hamiltonian,44 that can be obtained from a power Figure 4. (Color online) Excitonic energies for the lowest energy exciton states. The system has a C3vsymmetry with three representations: A1,Eand A2. The states 1 to 4 have Esymmetry and are valley degenerate; states 5 and 6 have A2and A1symmetries respectively and are non degenerate (see Ref. 33). series expansion of the tight-binding hamiltonian. The electron-electron interaction for a 2D material is given by the Keldysh potential.45 This effective model only considers the top valence band and the bottom conductance band. From the equation of motion we derive the following BSE: (ω−˜ωλk)pλ(k, ω)=(E0dλ(k) + Bkλ(ω)) ∆fk,(3) where λ=±,p±(k, ω)is the interband transition amplitude, ˜ωλkis the transition energy renormalized by the exchange self-energy and Bkλ(ω)is a term that renormalizes the Rabi-Frequency, dλ(k)is the dipole matrix element and ∆fkis the occupation difference, given by the Fermi-Dirac distribution. See Appendix Afor more details. From the homogeneous part of Eq. (3) we can obtain the exciton energies and the wave functions. Using the procedure explained in Ref. 43, we can obtain the corresponding Elliot formula for the optical conductivity: σ(ω) σ0 = 4i~ωX n pn ~ω−En+ iγ,(4) where nlabels the exciton state, γis the exciton linewidth, Enthe exciton energy, pnthe corresponding exciton weight and σ0=e2 4~. Fig. 6shows that the G0W0+BSE described in section III fits well to the Elliot formula, with a very good agreement in the real part and a small shift in the imaginary part. The energies and weights of the fit for the G0W0+BSE and the equation of motion method are compared in table VI. We use the parameters from Ref. 44:a0= 2.51 Å, t0= 2.33 eV ~vF=√3 2t0a0,2mv2 F= 3.92 eV. The Keldysh potential parameter r0was calculated in Ref. 33 to be r0= 10 Å. We can see a excellent agreement between the exciton energies of both methods. The difference in the weights pn 5 1 2 3 4 5 6 7 8 Figure 5. (Color online) Probability density |φ(re,rh)|2for the exciton states 1 to 8. The hole is localized slightly above the nitrogen atom (light color) at the centre of the lattice. 6 can be explained by the oversimplification of the Dirac hamiltonian used for the Elliot formula and consequently the less accurate dipole matrix elements that enter their calculation. Figure 6. (Color online) Fit of the Elliot formula to the G0W0+BSE result. There is a very good agreement for the real part and a small shift in the imaginary part; the exciton linewidth used was γ= 0.1eV. The parameters of the fitting are shown in table VI. Finally, we used the equation of motion to predict the behavior of the exciton energy and the K→K transition energy as a function of the environment dielectric constant. The result can be seen in Fig. 7. There is a strong decrease in the K→K transition energy and an almost linear behavior, also decreasing, of the first exciton energy as the external dielectric constant increases. This effect is simple to understand, since a large dielectric constant screens more effectively the electron-electron interaction. V. EXCITON-POLARITONS In this section we discuss the exciton-polariton modes in 2D hBN. Those modes are electromagnetic evanescent waves along the direction perpendicular to the hBN sheet. We assume that the hBN monolayer is cladded between two uniform, isotropic media with dielectric constants ε1and ε2and that the hBN sheet is in the xyplane. So the electromagnetic mode is evanescent in the z axis and proportional to e−κiz(i= 1,2). The modes can be classified as transverse magnetic or transverse electric (TM/TE). Table VI. Comparison of the Elliot formula parameters used in the G0W0+BSE calculation and the equation of motion approach. The spin and valley degeneracy is already included in the weight. E1(eV) p1E2(eV) p2 G0W0+BSE 5.48 0.088 6.41 0.027 Eq. of Motion 5.52 0.354 6.53 0.045 Figure 7. (Color online) Exciton and K→K transition energy as function of the environment dielectric constant. We can see that the dependence of the first exciton energy is almost linear while the K→K transition energy has a greater dependence on the dielectric constant. The dispersion relation for the TM mode is given by the solution given in Ref. 46: ε1 κ1 +ε2 κ2 + iσ(ω) ε0ω= 0,(5) and for the TE mode: κ1+κ2−iωµ0σ(ω) = 0,(6) with σ(ω)the hBN optical conductivity and: κi=rq2−εi ω2 c2,(7) where qis the exciton-polariton in-plane wavevector and cis the velocity of light in vacuum. We shall consider the simplest case of ε1=ε2= 1. A rule of thumb is that when =σ(ω)>0(=σ(ω)<0) TM (TE) modes are supported. A. Complex q×Complex ω First, we note that both Eqs. (5) and (6) are complex. Therefore, for a given q(ω) real, the solution will be a complex ω(q). Each of these approaches (complex qor complex ω) lead to different dispersion relations for the exciton-polaritons as discussed elsewhere.47–51 Both complex qand complex ωapproaches give the same results when an active media is used to balance the losses.51 The complex qapproach is suitable when the polariton is excited in a finite region of space with a monochromatic wave, while the complex ωapproach is valid instead when the entire sample is excited by a pulsed light.49 The dispersion relation for both the TE and TM modes in the complex ωapproach was obtained by solving Eqs. (5) and (6) and using the Elliot formula (4) with the parameters of table (VI) for the G0W0+BSE calculation and a damping of γ= 0.1eV. The result is shown in 7 Figure 8. (Color online) Exciton-polariton dispersion relation for complex frequency. The results are given as a function of the wavenumber ˜ν=λ−1 q. The gray dashed-dot line represents the light cone in air. In this approach, the wavenumber can reach large values for both TE and TM modes for either Aor Bexciton energies. Detail around excitons Aand Bis shown in the right panels. Fig. 8, where Aand Bdenote the first two excitonic energies. Both TE and TM modes can have a large localization (high κior q) in this case. The TE mode has a flat dispersion relation that approaches the exciton energy as qgoes to infinity. As expected, the TM mode has a higher frequency than the exciton energy while the TE mode has a lower frequency. We point out that, and contrary to graphene, the TE mode presents a high degree of localization. In the complex ωapproach both excitons Aand Bsupport polaritons. This can be understood by examining Eq. (4). As ~ωapproaches En−iγ, the corresponding contribution to the optical conductivity diverges. This quantity can be infinitely negative or positive depending on the real part of the frequency approaching En from the right or the left, supporting TM and TE modes respectively. Fig. 8also shows that the electrostatic limit qω/c is approached near both exciton energies. In that limit the lifetime τof the TM exciton-polariton τ=−1/=ωcan be calculated from (see Appendix B): τ−1=γ ~+1 ~ pn=[bn]  (ε1+ε2)bn 4παcq + 1 2,(8) where αis the fine-structure constant and bnis the contribution that arises from the background conductivity provenient from interband transitions and other excitonic states. For a negligible background bn≈0, the excitonpolariton lifetime is proportional to the inverse of the exciton linewidth γ. Next we shall consider the case of complex q. There will be then a simple relation to obtain qfor a given frequency (assuming εi= 1): c2q2=ω2+c2κ2 α(ω),(9) with α=TM/TE and from Eqs. (5) and (6) we have: κTE(ω)=i ε0ω 2σ(ω),(10a) κTM(ω)=iωµ0σ(ω) 2,(10b) The condition for the existence of polaritons is <κα>0. These equations allowed us to calculate the dispersion relation shown in Fig. 9for several values of the damping constant γ. The dependence of the γparameter of excitons was studied for WS2in Ref.52 as function of temperature, showing that the linewidth decreases as the temperature decreases. From Fig. 9we can see that the TE mode is strongly supressed except when the damping has the very low value of 4meV, close to the intrinsic line-width. The opposite happens for the TM mode, for which the dispersion relation is almost insensitive to the damping γ. An important figure of merit is the ratio of the propagation length `==q−1to the exciton wavelength λq= 2π/<q, as it indicates if a polariton can propagate before extinction, that is shown in Fig. 10 for several values of γ. The TM mode is highly supressed except for the very low γ= 4 meV, while the TE mode has higher propagation rate and two different qualitative behaviors. For larger γ, the propagation rate increases with the frequency while the opposite happens for γ= 4 meV. A better understanding of this behavior can be achieved if we consider the confinement ratio λ0/λq, with λ0being the wavelength of the free-radiation (see Figure 11). The confinement of the TM modes increases with increasing frequency and have a negligible γdependence. On the other hand, the TE modes are poorly confined, with the confinement going to zero faster with increasing γ. This explains the large propagation rate in this case: the poorly confined field is essentially attenuated free radiation, i.e., there are no more excitons being excited, but the radiation field is attenuated by the material free charges. The overall conclusion is that 2D hBN is a good platform for exciton-polaritons when we consider the complex ωapproach for both TM and TE modes. In the complex qapproach, the results show that exciton-polariton can be observed only for γ= 4 meV. B. UV radiation mirror It was pointed out recently that excitons in MoSe2 can lead to very high reflection of electromagnetic radiation53,54. In this section we show that the same 8 Figure 9. (Color online) Exciton-polariton dispersion relation in the complex wavenumber approach. Panel A (B) shows the TM (TE) mode. The TM mode has a dispersion almost insensitive to the relaxation rate while the TE mode changes significantly: the wavenumber is close to the free-light one and only for γ= 4 meV there is a different behavior. occurs with hBN, but in a different spectral range. We consider a free-standing hBN monolayer. In this case the reflection is given by:55 R= παf(ω) 2 + παf(ω) 2 ,(11) where f(ω) = σ(ω)/σ0,α≈137−1is the fine structure constant and σ0=e2/4~. Fig. 12 shows that the reflection can reach almost 100%for the value γ= 4 meV at the Aexciton energy. This is a consequence of the very high weights for hBN that appears in the Elliot formula (see table VI). We emphasize that those results are for a free-standing hBN sheet. The γvalue can be controlled by the temperature as discussed in the sections before. As shown in Fig. 7, the exciton energy and therefore the reflection peak can be controlled by varying the external dielectric constant. VI. CONCLUSION We calculated the band structure of 2D hexagonal boron nitride using DFT and the G0W0approximation. Figure 10. (Color online) Exciton-polariton propagation ratio. Panel A (B) shows the TM (TE) mode. The propagation rate of the TM mode is very low except for γ= 4 meV. The peak at ω= 5.48 corresponds to the propagation of radiation. As can be seen in Fig. 9, the wavenumber tends to the free-light wavenumber. The same result appears in the propagation rate for the TE modes: except for γ= 4 meV, all other modes correspond to poorly confined modes (see Fig. 11 also). For γ= 4 meV and the TE mode, the propagation rate decreases with the increasing frequency. Then the Bethe-Salpeter equation was used to determine the excitonic energies of hBN. We determined the values of the band gap, optical gap, excitonic binding energies using a first principles approach. The results are in very good agreement with the ones obtained using a very different approach, namelly the equation of motion formalism and the Elliot formula, which are also presented in this paper. This latter formalism allowed us to study the optical properties for both the TM and TE modes. Our results show that 2D hBN is a good candidate to polaritonics in the UV range. We also show that a single layer h-BN can act as an almost perfect mirror for ultraviolet electromagnetic radiation. ACKNOWLEDGMENTS R.M.R. and N.M.R.P. acknowledge support from the European Commission through the project “GrapheneDriven Revolutions in ICT and Beyond" (Ref. No. 785219), COMPETE2020, PORTUGAL2020, FEDER and the Portuguese Foundation for Science and Technol- 9 Figure 11. (Color online) Exciton-polariton confinement ratio. Panel A (B) shows the TM (TE) mode. The confinement of the TM mode increases with the frequency and has a small dependence with the relaxation rate γ. The TE modes for the higher values of γare poorly confined. For the value γ= 4 meV we have a peak in the confinement below the exciton energy. ogy (FCT) through project PTDC/FIS-NAN/3668/2014 and in the framework of the Strategic Financing UID/FIS/04650/2013. 1J. Bao, K. Jeppson, M. Edwards, Y. Fu, L. Ye, X. Lu, and J. Liu, Electronic Materials Letters 12, 1 (2016). 2J. Bao, M. Edwards, S. Huang, Y. 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