Cavity exciton—polaritons

Andreani, L. C., Gerace, D., Agio, M. (2005). Exciton-polaritons and nanoscale cavities in photonic crystal slabs. Physica Status Solidi (B), 242, 2197-2209. 

In Subsection 10.2.2, the Maxwell equations in an anisotropic microcavity are discussed and some important facts concerning Coulomb and mechanical excitons are summarized. In Subsection 10.2.3 the dispersion equation of cavity polaritons is derived. Its solutions for the cases of crystals with one and two molecules in the unit cell are discussed in Subsections 10.2.4 and 10.2.5, respectively. The main results are summarized in the conclusions. 

In this subsection we introduce some notations for the bare excitations whose interaction leads to the formation of cavity polariton states. These bare excitations are cavity photons and Coulomb excitons. 

Typical examples of the dispersion curves are shown in Fig. 10.2 for 99 = 7t/4, small positive detuning u = ivc — d>i = 35 meV, and different relations between Wy, W2 and Ac = 2 — uq. It is clear that when one of the coupling parameters IF is small, the spectrum consists of a doublet of polariton branches and of two branches which are close to the bare cavity photon and the bare exciton (Fig. 10.2a). When the Davydov splitting is small and W W2, then the pairs of the dispersion curves almost overlap (Fig. 10.2b). The electric fields in the overlapping curves are (with accuracy up to terms of the order of q2 IQ2) perpendicular to each other (see the discussion in Subsection 10.2.4). When all the parameters are of the same order, the spectrum consists of four well-pronounced polaritonic branches (Fig. 10.2c). 

The spectrum of the excitations is shown in Fig. 10.5 for 2 A = 80 meV. The dashed lines show the uncoupled molecular excitons and photons, and the solid lines show the coherent part of the spectrum with well-defined wavevector. The crosses show the end-points of the spectrum of excitations for which q is a good quantum number. The spectrum of incoherent (weakly coupled to light) states is shown by a broadened line centered at the energy Eq. It follows from the expression for the dielectric tensor that this spectrum is the same as the spectrum of out-of-cavity organics. The spectrum of absorption as well as the dielectric tensor depend on temperature. This means that in the calculation of the temperature dependence of the polariton spectrum we have to use the temperature dependence of the resonance frequency Eo as well as the temperature dependence of 7 determining the width of the absorption maximum. However, the spectrum of emission of local states which pump polariton states can be different from the spectrum of absorption. The Stokes shift in many cases... 

FlG. 10.5. The dispersion curves of the coherent polaritonic states (solid lines) and of uncoupled cavity photons and the molecular excitation (dashed lines). The crosses show the end-points of the part of the spectrum with well-defined wavevector. On the right, the broadened line of the molecular resonance is shown. The inset shows the excitonic weights (10.26) for upper ( cix ) 2) and lower ( cix ) 2) polaritonic branches. Reprinted with permission from Agranovich et al. (15). Copyright 2003, American Physical Society. 

The quantity N 2 gives us the Wannier-Mott exciton component, A f 2 the part of the Frenkel exciton, and Np 2 the part of the cavity photon mode in the (p,k) cavity polariton state with energy hup(k). The diagonal Hamiltonian is... 

Operator (13.113) has terms with p p which give us the transitions from one branch p to another p. It is seen from eqn (13.114) that in the region of strong renormalization of cavity modes, the polariton-phonon interaction will be determined by the structure of the p, p modes. For example, to have a large contribution of the exciton-phonon interaction from organic QWs to the transition p —> p with emission of a phonon we need large enough values of NF (7, ) and NF(k — q) 2. 

FlG. 13.15. (a) Dispersion curve for cavity polariton and energy relaxation of Wan-nier-Mott excitons (b) coupling coefficient of the lower branch to cavity photon and to exciton. 

Figure 8.18 (a) Angle-resolved PL spectra at RT in the range of 0 -40° for a X-thick ZnO hybrid MC. The dotted line is the exciton mode. The solid lines are guides to the eye. (b) Experimental cavity polariton dispersion curve. The dashed lines represent the cavity and exciton modes. (Courtesy of R. Shimada [140].)... 

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