Surface Phonon Polaritons

Let us turn now to the case when light is coupled with optical phonons in an ionic crystal. The corresponding dielectric function can be written in two equivalent forms 

The condition (3.90) for the existence of SPs is then fulfilled in fhe frequency range 

We have seen that SPs are nonradiative modes and that they caimot be excited by incident light at a flat interface between two media. This follows from 

One can elongate the wave vector of light by passing it through a medium (a prism) with a refractive index n Otto configuration (Otto 1968)) (see Fig. 3.5a). In this case, the wave vector component kx does not change across both interfaces, prism/medium 1 and medium 1/medium 2, and is given by 

In medium 1, the wave vector component perpendicular to the interface is found as 

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Figure 3.7. Sketch of surface phonon-polariton at air-quartz interface in xy-plane propagating as damped wave in x-direction v = 1110 cm. ...

Depending on the substrate excitations which are coupled with light into the SP mode, one distinguishes surface plasmon polaritons, surface phonon polaritons, surface exciton polaritons, etc. In this section we shall consider surface plasmon polaritons in some detail. This type of electromagnetic wave was first discussed by Sommerfeld in connection with the propagation of radiowaves along the Earth s surface (Sommerfeld 1909). 

In Section 3.3.3 we have considered surface phonon polaritons in dielectrics and semiconductors. To derive their dispersion relation we have used the dielectric function of a crystal in the form (3.96). However, for comparison with experimental data it is necessary to take into account the decay rate of phonons, F(a ), as well as their anharmonicity characterized by the frequency shift A(ur) (Mirlin 1982). The corresponding dielectric function is given by... 

Problem 4.5. Neglecting anharmonicity of phonons, obtain an expression for the propagation length of surface phonon polariton at a crystal-vacuum interface at frequencies close to (Vjo in terms of the decay rate of phonons, r [Pg.109]

Mirlin, D. (1982). Surface phonon polaritons in dielectrics and semiconductors. In Agranovich, V. and Mills, D., editors. 

Figure 8.2.19 Dispersion of the transversal and longitudinal phonon branches (solid blue lines) for small wave vectors. The asymptotic linear behaviors for fc and /t > -j, respectively, are outline by dashed black lines. The dispersion of the surface phonon-polariton, which exists in the gap of the bulk transversal and longitudinal phonon branches, is shown as a red line. Asymptotically, it converges for larger wave vectors to the constant frequency a). ...

Figure 8.2.20 Snapshot of the electrical field lines E x,z) above the solid surface (side view) for a surface phonon-polariton (Fuchs-Kliewer phonon) with wavelength X. Blue and red circles emphasize the positive and negative sign of surface charges.

Ions in the lattice of a solid can also partake in a collective oscillation which, when quantized, is called a phonon. Again, as with plasmons, the presence of a boundary can modify the characteristics of such lattice vibrations. Thus, the infrared surface modes that we discussed previously are sometimes called surface phonons. Such surface phonons in ionic crystals have been clearly discussed in a landmark paper by Ruppin and Englman (1970), who distinguish between polariton and pure phonon modes. In the classical language of Chapter 4 a polariton mode is merely a normal mode where no restriction is made on the size of the sphere pure phonon modes come about when the sphere is sufficiently small that retardation effects can be neglected. In the language of elementary excitations a polariton is a kind of hybrid excitation that exhibits mixed photon and phonon behavior. 

We note that ionic crystals may have dielectric functions satisfying Eq. (4) for frequencies between their transverse and longitudinal optic phonon frequencies. SEW on such crystals are often called surface phonons or surface polaritons and the frequency range is the far IR. 

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.

It is expected that the above method will be applied to the study of propagation and relaxation of phonon-polaritons in a wide variety of non-centrosymmetric crystals. The technique also allow a direct investigation of the interaction of the polariton with spatial inhomogeneities in the crystal (such as the crystal surface which is also a new field of study, and should give access to other important problems in vibrational dynamics, such as the interaction between phonon-polariton beams in crystals. 

When the frequency of the surface biphonon lies within the band of the surface polariton, Fermi resonance occurs and the dispersion curve of the po-lariton is subject to a number of essential changes (gaps appear, etc. (86)). Consequently, experimental research of surface polariton dispersion under these conditions could yield, like similar investigations of bulk polaritons, a great deal of interesting information, not only about the surface biphonons themselves, but about the density of states of surface phonons and the magnitude of their anharmonicity constants as well. 

In real systems, both ez and kx are complex. As a consequence, the resonance condition (3.12) becomes ei - - 2 = min. Moreover, the surface mode propagating along an interface has a finite length Lx = 1/ (of the order of 10-50 p,m for dielectrics). To illustrate this, a surface phonon at the air-quartz surface is depicted in Fig. 3.7. In both media, the penetration depth of the surface polariton depends on the permittivity for example, at the air-quartz interface at 1110 cm the penetration depths are d 3.5 p,m and d z 0.56 p,m in air and quartz, respectively. 

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). 

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 ... 

Surface biphonons could be investigated, for example, by the attenuated total reflection (ATR) method. In contrast to RSL by polaritons, this method is effective, as is well known, both for crystals with and without inversion center. In this sense, it is a more universal method. In conclusion we point out that in degenerate semiconductors Fermi resonance with plasmons (47) is also possible along with Fermi resonance with phonons and polaritons. The spectrum of plasmophonons has been measured in many semiconductors by the RSL method (see, e.g. Mooradian and McWhorter (48)). 

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