Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant

The evaluation of the amount of tetrahedral and trigonal cross-linking, that is, the ${\mathit{sp}}^{3}$- and ${\mathit{sp}}^{2}$-hybridized carbon, is of great importance in understanding the properties of amorphous carbon films. In this paper we report a method for deducing the [${\mathit{sp}}^{3}$]/[${\mathit{sp}}^{2}$] ratio from the experimental values of the complex dielectric constant as obtained by optical transmittance and reflectance measurements. We assume a Gaussian-like distribution of \ensuremath{\pi} and ${\mathrm{\ensuremath{\pi}}}^{\mathrm{*}}$ electronic densities of states in order to fit the contribution of \ensuremath{\pi}\ensuremath{\rightarrow}${\mathrm{\ensuremath{\pi}}}^{\mathrm{*}}$ to the imaginary part, ${\mathrm{\ensuremath{\epsilon}}}_{2}$, of the dielectric constant in the low-energy region. Through the Kramers-Kronig relationships we deduce the corresponding values of the real part ${\mathrm{\ensuremath{\epsilon}}}_{1}$ of the dielectric constant for such transitions. By subtracting these values from the measured ${\mathrm{\ensuremath{\epsilon}}}_{1}$ we deduce the contribution of \ensuremath{\sigma}\ensuremath{\rightarrow}${\mathrm{\ensuremath{\sigma}}}^{\mathrm{*}}$ to ${\mathrm{\ensuremath{\epsilon}}}_{1}$. The Wemple-Didomenico model has been used to obtain the dispersion energy and the average excitation energy. Knowing the plasmon energies, we apply the ``f-sum rule'' to deduce the [${\mathit{sp}}^{3}$]/[${\mathit{sp}}^{2}$] ratio. The method applied to a-C:H films deposited by rf diode sputtering provides results in agreement with those obtained by other techniques.