Electron energy loss in ordered arrays of polarizable spheres Carlos I. Mendoza and Rube ´ n G. Barrera Instituto de Fı ´sica, Universidad Nacional Auto ´noma de Mexico, Apartado Postal 20-364, 01000 Mexico Distrito Federal, Mexico Ronald Fuchs Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011 Received 9 June 1998; revised manuscript received 14 June 1999 We develop a theory for the energy loss of swift electrons traveling parallel to an ordered array of polariz- able spheres. The energy loss is given in terms of a surface response function which is expressed as a spectral representation. The poles and weights in this representation are determined through the eigenvalues and eigenvectors of an interaction matrix. This matrix takes account of the quasistatic electromagnetic interaction between the polarized spheres to an arbitrary multipolar order. We use our theory to calculate the energy-loss spectra for cubic arrays of aluminum spheres with various numbers of layers and compare the results with those obtained using a dielectric continuum model. S0163-18299905843-9 I. INTRODUCTION Electron-energy-loss spectroscopy EELS of inhomoge- neous systems has been an active field of research during the last decades. Here, we will be interested in the calculation and analysis of EELS spectra of granular matter. The calcu- lation of the energy loss of swift electrons passing through a system of nanometric inclusions embedded in an otherwise homogeneous matrix was stimulated by the recent experi- ments of Walsh. 1 The concept of an effective medium for the calculation of the energy-loss function in a granular compos- ite has been very appealing because one might expect that this function could be written in terms of the effective dielec- tric function associated with the composite. The first at- tempts along these lines were done by using the effective dielectric functions which had proved to be successful in describing the optical properties of granular composites, 1 like the ones devised, for example, by Maxwell Garnett, 2 Bruggeman, 3 or Landau and Lifshitz. 4 The main problem encountered in using these types of effective dielectric func- tions was that the peaks in energy loss coming from the excitation of the bulk plasmons of the inclusions did not appear in the calculated spectra. The origin of this problem was the local nature of the effective dielectric response, that is, the effective dielectric response depended only on the frequency of the applied field and had no dependence on its wave vector. This actually means that the response is valid only in the limit as the wave vector tends to zero. Although this limit might be appropriate when the system interacts with light, this is certainly not true when the applied field is the field carried by a moving electron, as in the case of EELS. One would expect that an effective dielectric function that could describe properly the energy-loss process should be nonlocal, that is, should depend on the frequency and the wave vector of the applied field. This approach was taken by Barrera and Fuchs, 5 who find a nonlocal effective dielectric response that could be used to calculate the energy-loss spec- tra of fast electrons passing through a system of random spherical inclusions contained in a matrix. In their approach, it was assumed that both the spheres and the matrix were described by local dielectric responses but the interaction among the polarized spheres was taken to all multipolar or- ders within the mean-field approximation. The calculated spectra using this theory showed well-defined peaks coming from the excitation of the bulk plasmons of the inclusions and the matrix, as well as the ones coming from the excita- tion of interfacial modes, that is, modes in which the induced charge is located at the interface of the spheres and the ma- trix. These calculated spectra also agreed with the experi- mental spectra of Walsh. Further theoretical developments 6 also showed the merits and limitations of an ad hoc phenom- enological theory 7 devised to explain the experimental re- sults. These developments have also shown the possibility of defining an effective local dielectric response that could de- scribe the energy-loss process. There is also interest in the calculation of energy-loss spectra for an experimental setup in which the electron trav- els parallel to the surface of the sample. Since there are cases in which it is actually not possible to construct very thin samples, one of the advantages of this experimental setup is that the electron does not have to go through the sample. Nevertheless, there is also the question of how much infor- mation about the surface structure will be contained in these energy-loss spectra. Answers to this question have been pro- vided using different approaches. For example, the authors of Ref. 8 have extended to a half space the idea of a nonlocal effective dielectric response discussed above for a system of random spherical inclusions. In order to do this they used a simple model for the structure of the interface together with an ad hoc elimination of nonphysical features in the energy- loss spectrum. On the other hand, Pendry and Martı ´ n-Moreno 9 PMM devised a calculation procedure to obtain the energy-loss spectra of fast electrons traveling par- allel to a half space or a slab occupied by an ordered sys- tem of spheres. In this procedure, the fields are decomposed on a transverse basis and the reflection coefficients of the half space are found by a finite-element numerical technique. The energy loss of the electron traveling along a rectilinear classical trajectory above the half space, or a finite slab, is calculated in terms of these reflection coefficients. The cal- PHYSICAL REVIEW B 15 NOVEMBER 1999-I VOLUME 60, NUMBER 19 PRB 60 0163-1829/99/6019/1383115/$15.00 13 831 ©1999 The American Physical Society