Coulomb surface excitons

Since the irreducible representations of the group of translations are onedimensional and are determined by the given wavevector, among the set of quantum numbers characterizing a surface exciton there is always a quasicontinuous quantum number - the wavevector. This vector differs from the corresponding one in the case of bulk excitons it may be directed only along the crystal surface and assumes values only within a two-dimensional Brillouin zone. In 

One of the main theoretical problems is to determine the dependence of the energy of a surface exciton on its wavevector or, in other words, to obtain the dispersion law for surface excitons. Then the next problems arise in consideration of the interactions of these excitons with light, phonons and with surface defects. 

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As a matter of fact, as can be seen from eqn (12.17), the limit k —> oo is consistent with the sum t (u>) + e2(w) approaching zero, which in the particular case of the boundary with vacuum (ei = 1, e2 = e) agrees with eqn (12.12). The result of this limiting transition confirms once more the remark made above, viz. that surface polaritons for large values of k transfrom into Coulomb surface excitons. The dispersion law for Coulomb surface excitons at a sharp boundary and without taking spatial dispersion into account has the form... 

As stressed earlier, the transition from surface polaritons to Coulomb surface excitons corresponds to the limiting transition c —> oo. For p-polarized waves it yields the dispersion relation... 

The macroscopical surface excitons obtained when retardation is taken into account, i.e. surface polaritons, cannot spontaneously transform into bulk emitted photons. Therefore, surface polaritons are sometimes said to have zero radiation width (it goes without saying that a plane boundary without defects it implies). At the same time the Coulomb surface excitons and polaritons in two-dimensional crystals possess, as was shown in Ch. 4, the radiation width T To(A/27ra)2, where A is the radiation wavelength, a is the lattice constant, and To the radiative width in an isolated molecule. For example, for A=500 nm and a = 0.5 nm the factor (A/2-7Ta)2 2x 104, which leads to enormous increase of the radiative width. For dipole allowed transitions To 5x10 " em, so that the value of T 10 cm-1 corresponds to picosecond lifetimes r = 2-kK/T x, 10 12s. 

Equation (12.78) implies that the frequency of a Coulomb surface exciton satisfies the equation... 

For electronic transitions in a dielectric the frequencies of Coulomb surface excitons are in the region of the longitudinal-transverse splitting, which is always small in comparison with the frequencies uy. Taking this fact into account, eqn (12.80) can be changed into... 

One of the most obvious perturbations of the surface excitons by the bulk crystal is the short-range coulombic interactions causing surface-exciton transfer to the bulk. We discard these perturbations on the grounds both of theoretical calculations27 and of the experimental observations, which show the presence of a surface exciton (the second subsurface exciton S3 see Fig. 3.2) resolved at about 2cm-1 above the bulk-exciton resonance. 

As illustrated in Fig. 2.8 of Section II, the general reflectivity lineshape shows (1) a sharp rise of the bulk 0-0 reflectivity (Section II.B.C) at E00, corresponding to the b coulombic exciton with a wave vector perpendicular to the (001) face (2) a dip, corresponding to the fission in the surface of a bulk polariton into one 46 -cm 1 phonon and one b exciton at E°° + 46 cm"1 (3) two vibrons E200 and E1 00 immersed in their two-particle-state continua with sharp low-energy thresholds. On this relatively smooth bulk reflectivity lineshape are superimposed sharp and narrow surface 0-0 transition structures whose observation requires the following ... 

Phenomenological theory of surface Coulomb excitons and polaritons... 

In a bulk semiconductor, photoexcitation generates electron-hole pairs which are weakly bounded by Coulomb interaction (called an exciton). Usually one can observe the absorption band of an exciton only at low temperature since the thermal energy at room temperature is large enough to break up the exciton. When the exciton is confined in an energy potential, the dissociation probability reduces and the overlap of the electron and hole wavefunction increases, which is manifested by a sharper absorption band observable at room temperature. This potential can be due to either a deformation in the lattice caused by an impurity atom or, in the present case, the surface boundary of a nanocluster. The confinement of an exciton by an impurity potential (called bound exciton) is well known in the semiconductor literature [16]. There is considerable similarity in the basic physics between confinement by an impurity potential and confinement by physical dimension. The confinement effects on the absorption cross section of a nanocluster are discussed in Section II. 

The photophysical processes of semiconductor nanoclusters are discussed in this section. The absorption of a photon by a semiconductor cluster creates an electron-hole pair bounded by Coulomb interaction, generally referred to as an exciton. The peak of the exciton emission band should overlap with the peak of the absorption band, that is, the Franck-Condon shift should be small or absent. The exciton can decay either nonradiatively or radiative-ly. The excitation can also be trapped by various impurities states (Figure 10). If the impurity atom replaces one of the constituent atoms of the crystal and provides the crystal with additional electrons, then the impurity is a donor. If the impurity atom provides less electrons than the atom it replaces, it is an acceptor. When the impurity is lodged in an interstitial position, it acts as a donor. A missing atom in the crystal results in a vacancy which deprives the crystal of electrons and makes the vacancy an acceptor. In a nanocluster, there may be intrinsic surface states which can act as either donors or acceptors. Radiative transitions can occur from these impurity states, as shown in Figure 10. The spectral position of the defect-related emission band usually shows significant red-shift from the exciton absorption band. 

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