Optical Properties of Aluminum

Abstract

The frequency-dependent complex dielectric constant $\ensuremath{\epsilon}(\ensuremath{\omega})={\ensuremath{\epsilon}}_{1}+i{\ensuremath{\epsilon}}_{2}$ and associated functions are derived in the range 0 to 22 eV by application of the Kramers-Kronig relations to existing reflectance data for clean Al surfaces. The results are quantitatively interpreted in terms of intra- and interband transitions as well as plasma oscillations. The decomposition of $\ensuremath{\epsilon}(\ensuremath{\omega})$ into intra- and interband parts given here is seen to be valid in the presence of electron-electron interactions. Due to these interactions the optical effective mass ${m}_{a}=1.5$, deduced from experiment in the free-carrier region, is appreciably larger than that obtained using Segall's band calculations (${m}_{\mathrm{ac}}\ensuremath{\cong}1.15$). The band calculations are extended to higher energies in order to examine the effect of interband transitions for the range of interest. It is found that the only interband transitions which lead to significant structure in ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ are those that occur around $W$ and $\ensuremath{\Sigma}$ in the vicinity of $K$ in the Brillouin zone and that these produce a peak near 1.4 eV. These conclusions are in accord with the experimentally determined ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ which exhibits a peak at 1.5 eV and has no further structure at higher energies. The result of a quantitative calculation of the structure in ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ using a fine mesh of points in k space and an approximate variation of the momentum matrix element with k is in good agreement with the experimental results with respect to shape but has a magnitude which is somewhat too low. From the known influence of many-electron effects on the intraband contribution to $\ensuremath{\epsilon}(\ensuremath{\omega})$ and a general sum rule, the corresponding effect on interband transitions may be estimated and shown roughly to account for the difference. The derived $\ensuremath{\epsilon}(\ensuremath{\omega})$ indicates the presence of a sharp plasma resonance at $\ensuremath{\hbar}{\ensuremath{\omega}}_{p}=15.2$ eV, in excellent agreement with the results of characteristic energy loss experiments. It is shown that this resonance may be interpreted either in terms of electrons characterized by the low-frequency optical mass and screened by the interband dielectric constant at ${\ensuremath{\omega}}_{p}$ or, since the $f$ sum has been essentially exhausted, in terms of the exact asymptotic formula for $\ensuremath{\epsilon}$ in which all carriers are unscreened and have the free-electron mass.