Density,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>sp</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>fraction, and cross-sectional structure of amorphous carbon films determined by x-ray reflectivity and electron energy-loss spectroscopy

Grazing-angle x-ray reflectivity (XRR) is described as an efficient, nondestructive, parameter-free means to measure the mass density of various types of amorphous carbon films down to the nanometer thickness range. It is shown how XRR can also detect layering if it is present in the films, in which case the reflectivity profile must be modeled to derive the density. The mass density can also be derived from the valence electron density via the plasmon energy, which is measured by electron energy-loss spectroscopy (EELS). We formally define an interband effective electron mass ${m}^{*},$ which accounts for the finite band gap. Comparison of XRR and EELS densities allows us to fit an average ${m}^{*}=0.87m$ for carbon systems, m being the free-electron mass. We show that, within the Drude-Lorentz model of the optical spectrum, ${m}^{*}=[1\ensuremath{-}{n(0)}^{\ensuremath{-}2}]m,$ where $n(0)$ is the refractive index at zero optical frequency. The fraction of ${\mathrm{sp}}^{2}$ bonding is derived from the carbon K-edge EELS spectrum, and it is shown how a choice of ``magic'' incidence and collection angles in the scanning transmission electron microscope can give ${\mathrm{sp}}^{2}$ fraction values that are independent of sample orientation or anisotropy. We thus give a general relationship between mass density and ${\mathrm{sp}}^{3}$ content for carbon films.

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