Electron-energy loss spectroscopy of multilayered materials: Theoretical aspects and study of interface optical phonons in semiconductor superlattices

The dielectric theory of electron-energy-loss spectroscopy (EELS) in the reflection geometry is reformulated so as to allow for an arbitrary compositional variation of the target material in the direction perpendicular to its surface. A new, general expression for the energy-loss spectrum is obtained in terms of an exact ‘‘effective dielectric function.’’ It is shown that the effective dielectric function, valid at long wavelength, can be obtained by solving a Riccati differential equation, the only information required being the otherwise arbitrary profile, in the normal direction z, of the dielectric constant ε(ω,z) for the frequency region of interest. Attention is then paid to an idealized multilayered material, made of an arbitrary succession of homogeneous layers separated by sharp interfaces parallel to the free surface. If ε(ω,z) is assumed to take constant values within each layer, an original solution is obtained for the Riccati equation, giving the effective dielectric function of the material in the form of a continued fraction. For the thick periodic multilayers or superlattices of current interest, the continued fraction can be evaluated analytically. Applications will be presented for the phonon EELS spectrum of polar-semiconductor superlattices for which a Lorentzian model for the ω dependence of the infrared dielectric constant can be used. Two kinds of vibrational excitations are theoretically predicted in such superlattices: (i) Bloch-like, Fuchs-Kliewer interface modes, propagating throughout the layers; and (ii) evanescent, Fuchs-Kliewer surface or interface modes. It is shown that the Bloch modes are responsible for weak EELS continua, while the localized modes give rise to strong peaks in the spectrum. Furthermore, it is predicted that the main features of the EELS spectrum sensitively depend on the relative thicknesses of the two alternating layers of the superlattice.