Surface excitons and polaritons

The elementary surface excited states of electrons in crystals are called surface excitons. Their existence is due solely to the presence of crystal boundaries. Surface excitons, in this sense, are quite analogous to Rayleigh surface waves in elasticity theory and to Tamm states of electrons in a bounded crystal. Increasing interest in surface excitons is provided by the new methods for the experimental investigation of excited states of the surfaces of metals, semiconductors and dielectrics, of thin films on substrates and other laminated media, and by the extensive potentialities of surface physics in scientific instrument making and technology. 

In the experimental study of surface excitons various optical methods have been used successfully, including the methods of linear and nonlinear spectroscopy of surface polaritons. A particularly large body of information has been obtained by the method of attenuated total reflection of light (ATR), introduced by Otto (1 2) (Fig. 12.1) to study surface plasmons in metals. Later the useful modification of ATR method also was introduced by Kretschmann (3) (the so-called Kretschmann configuration, see Fig. 12.2). The different modification of ATR method has opened the way to an important development in the optical studies of surface waves and later was used by numerous authors for investigations of various surface excitations. 

Experiments have also been started that use the inelastic light scattering and include the methods of coherent anti-Stokes Raman spectroscopy (CARS), as well as electron energy loss spectroscopy (EELS). The methods (see, for instance, (4)(5)) are based on the application of various physical processes, as can be seen from their names. Accordingly they complement one another and enable us to study the elementary excitations of a surface over a wide range of energies and wave vectors. 

In this section we mainly discuss the theoretical aspects of surface excitons. In this connection we should emphasize, first of all, that the determination of the wavefunctions and energies of surface excitons for different crystal models in general requires the application of microtheoretical methods. An exception is the case when the thickness of the subsurface layer L in which the surface excitons are localized substantially exceeds the lattice constant of the crystal a. Such surface states can be investigated within the framework of macroscopic phenomenological electrodynamics, which uses one or another model of the crys- 

Experimental Otto configuration for (a) single prism ATR measurements and (b) double prism measurement of propagation length L. Prism are spaced a variable distance g above surface (from (6)). 

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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. 

Macroscopic surface excitons and polaritons in isotopically mixed crystalline solutions... 

A theory of 2D excitons and polaritons is presented for this type of surfaces, with continuity conditions matching 2D states their 3D counterparts in the bulk substrate, investigated in Sections I and II. This leads to a satisfactory description of the excitations (polaritons, excitons, phonons) and their theoretical interactions in a general type of real finite crystals A crystal of layered structure (easy cleavage) with strong dipolar transitions (triplet states do not build up long-lived polaritons). 

Phenomenological theory of surface Coulomb excitons and polaritons... 

To explain the observed width, it is necessary to look for strong surface-to-bulk interactions, i.e. large magnitudes of surface-exciton wave vectors. Such states, in our experimental conditions, may arise from virtual interactions with the surface polariton branch, which contains the whole branch of K vectors. We propose the following indirect mechanism for the surface-to-bulk transfer The surface exciton, K = 0, is scattered, with creation of a virtual surface phonon, to a surface polariton (K / 0). For K 0, the dipole sums for the interaction between surface and bulk layers may be very important (a few hundred reciprocal centimeters). Through this interaction the surface exciton penetrates deeply into the bulk, where the energy relaxes by the creation of bulk phonons. The probability of such a process is determined by the diagram... 

Figure 3.15. Diagram of a nonlocal surface-exciton transfer, corresponding to the optical creation of a surface exciton followed by its relaxation to the bulk. The essential virtual stage is the scattering of a surface phonon (K 0) and the creation of a surface polariton with a large wave vector (K 0), producing large interaction energies with the bulk. 21 Then relaxation in the bulk is ultrafast.

The model of an isolated layer was refined by introducing substrate effects by coupling the surface 2D excitons to the bulk polaritons with coherent effects modulating the surface emission and incoherent k-dependent effects damping the surface reflectivity and emission, both effects being treated by a KK analysis of the bulk reflectivity. The excitation spectra of the surface emission allowed a detailed analysis of the intrasurface relaxation dominated by resonant Raman scattering, by vibron fission, and by nonlocal transfer of... 

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... 

If the thickness of the surface layer in which a surface exciton-polariton is localized considerably exceeds the lattice constant of a crystal, the electric and magnetic field strength vectors, i.e. vectors E and H of a wave with energy hw in both media (in vacuum and in the crystal the crystal is assumed to be nonmagnetic so that the magnetic induction vector B = H), satisfy Maxwell s equations... 

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... 

In the preceding derivation of the frequencies of surface polaritons and surface excitons the boundary conditions were applied at a sharp boundary without surface currents and charges. In this simplest version of the theory the so-called transition subsurface layer has been ignored however, this layer is always present at the interface between two media, and its dielectric properties differ from the dielectric properties of the bulk. Transition layers may be of various origins, even created artificially, e.g. by means of particular treatment of surfaces or by deposition of thin films of thickness dphenomenological theory it is rather easy to take account of their effects on surface wave spectra in an approximation linear in k (15). 

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 ... 

Photoluminescence could be due to the radiative annihilation (or recombination) of excitons to produce a free exciton peak or due to recombination of an exciton bound to a donor or acceptor impurity (neutral or charged) in the semiconductor. The free exciton spectrum generally represents the product of the polariton distribution function and the transmission coefficient of polaritons at the sample surface. Bound exciton emission involves interaction between bound charges and phonons, leading to the appearance of phonon side bands. The above-mentioned electronic properties exhibit quantum size effect in the nanometric size regime when the crystallite size becomes comparable to the Bohr radius, qb- The basic physics of this effect is contained in the equation for confinement energy,... 

Takeda E, Nakamura T, Fujii M, Miura S, Hayashi S (2006) Surface plasmon polariton mediated photoluminescence from excitons in silicon nanocrystals. Appl Phys Lett 89(10) 101907 Tsybeskov L, Duttagupta SP, Fauchet PM (1995) Photoluminescence and electroluminescence in partially oxidized porous silicon. Solid State Commun 95(7) 429-433 Tsybeskov L, Duttagupta SP, Hirschman KD, Fauchet PM (1996) Stable and efficient electroluminescence from a porous silicon-based bipolar device. Appl Phys Lett 68(15) 2058-2060 Valenta J, Lalic N, Linnros J (2004) Electroluminescence of single silicon nanocrystals. Appl Phys Lett 84(9) 1459-1461... 

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