The momentum and energy dependence of the matrix elements for direct and indirect transitions across the band gap is studied both theoretically and experimentally. The accepted theory for the inelastic scattering cross section of fast electrons by condensed matter is extended to show how the nature of a transition can change the shape of the measured energy-loss spectrum in the region of the onset. For the case of direct transitions the matrix element acts only as a multiplicative factor to the joint density of states (JDOS), and so an $(E-E_g{)}^{1/2}$ term is observed in the spectrum. The matrix elements for indirect transitions are shown to be dependent on the momentum transferred by the incident fast electron to the crystal electrons. The product of the indirect matrix element and the JDOS contributes an $(E-E_g{)}^{3/2}$ term to the spectrum. To test this theory it is shown that the matrix-element-weighted joint density of states should be extracted from the electron-energy-loss spectra via a Kramers-Kronig transformation. An objective method is proposed for plotting the data to determine both the principal band gaps and their direct or indirect nature. The method is tested and succeeds in well-known cases. It is now possible with the electron microscope to measure these fundamental electronic properties of semiconductors and insulators accurately by electron spectroscopy.

• Received 17 December 1997

DOI:https://doi.org/10.1103/PhysRevB.58.10326