Exciton dispersion

Figure 1.2. Excitonic dispersion curves for the first singlet state of the anthracene crystal. These curves are calculated in the point-dipole approximation, the transitions to upper states being accounted for by a constant dielectric permittivity.13,37...

Figure 2.7. The various responses of the 0-1 vibron-two-particle states described by the hamiltonian (2.65H2.69) vs the two parameters EFC and AO, for a 2D lattice with two molecules per cell. The scheme of the absorption is given in each delimited area. Apart from the well-known case FC = hO0 ( 2 = 1), the boundaries of the areas depend on the excitonic dispersion of the lattice. Rigorously speaking, this scheme is valid only for a 3D lattice.

Exciton dispersion As the 0-0 exciton dispersion in anthracene is somewhat complex,101 we assumed a bidimensional parabolic dispersion (effective-mass approximation). This assumption is justified by the absence of interplane coupling for k perpendicular to plane (001) planes2 9—the case of b-polarized exciton under study—and it is aimed to give back the observed stepwise thresholds bound to the bottom of the band. The model most likely fails for the exciton band as a whole, but anyhow, we could not avoid it without lengthening computation times excessively. 

Figure 2.18. Profile of isoenergetic surfaces of the excitonic dispersion in the vicinity of the bottom of the band in the model (2.139) of an orthorhombic crystal. We note the lengthening of the surfaces along c . Also, the wave vectors tend to orient perpendicular to d (or the b axis) in the vicinity of the point K = 0.

The modification of the surface excitonic dispersion is inexplicable in the absence of reconstruction. 

The plane waves of a perfect 2D lattice diagonalize the electromagnetic interactions, giving rise to the excitonic dispersion through the Brillouin zone, and to the surface-exciton-polariton phenomenon around the zone center.148,126 The corresponding hamiltonian may be written as... 

This dependence of the dressed exciton dispersion ( k) for angle 9 = 0 when the transition dipole moment is perpendicular to chain is displayed in Fig. 4.4. For another orientation of the exciton transition dipole moment the dependence of k) can be very different. For excitons with small transition dipole moment the renormalization of the exciton dispersion due to account of retardation is usually small and can be important only at low temperature of order of 1-2 K or less because of the smallness of the parameter A/Efx. In the same situation the radiative width of exciton states with small wavevectors determined by the same parameter A/EM can be a hundred-fold larger than the radiative width of a molecule in solution. Very interesting is the problem of the temperature dependence of the radiative lifetime and we come back to the discussion of this problem later. 

Quite analogously we can consider retardation effects in two-dimensional crystals. In this case the excitonic dispersion rule has the form... 

Fig. 4.5. Excitonic dispersion in two-dimensional crystals with retardation.

Equation (4.109) applies if < ". The lower branch (a < 0) satisfies this condition since there " = 0. At the upper branch (a > 0) eqn (4.113) applies only for those values of k for which the quantity defined in (4.108) is much smaller than (k). This condition determines the limit of the upper branch. In Fig. 4.5, where the excitonic dispersion in two-dimensional crystals with retardation accounted for is schematically sketched, the limit is indicated by an asterisk. 

The appearance of an enhanced radiative width and renormalization of the exciton dispersion are the main effects arising in one- and two-dimensional structures under the influence of retardation (21). Qualitatively these effects are valid for Frenkel as well as for Wannier-Mott excitons. In contrast to 3D structure where in the exciton-photon interaction all three components of the momentum have to be conserved and as a result a picture with 3D polaritons arises, for structures of lower dimensionality only the in-plane momentum for 2D structures is conserved and only one component for the ID structure. An exciton in both cases is coupled to a continuum of photon states. There is no possibility of reversible strong... 

In the two-particle picture (Fig. 1), the optical absorption by excitons involves the conversion of a photon into an exciton, the absorption occurring at a place where phonon dispersion curve intersects the exciton dispersion curve, meeting... 

To compute the complete exciton dispersion relations E(k), we start from Eqns. (6.16)-(6.18). Denoting the two possible sites within the unit cell by A and B, we write the fe-dependent part of the energy as ... 

X Y(T) impurity or dopant concentration Yoshida function O) (q) frequency magnetic exciton dispersion... 

Here x (q, co) and the single ion dynamical susceptibility u( >) are tensors in both sublattice (A, B) and transverse xy) Cartesian coordinates. The poles of eq. (84) determine the collective excitations of 5f-local moments. In the paramagnetic phase they are given by the magnetic exciton dispersion... 

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