=================================================================================
Band gap can be determined based on Bethe’s theory. In this case, the SSD (single scattering distribution) spectrum is described to be proportional to the product of the joint density of states (JDOS). Here, the JDOS is given by,
for a direct band gap,  [2405a]
and,
for an indirect band gap  [2405b]
where,
I_{0} and c  The constants.
E  The energy loss.
Therefore, E_{g} can be extracted by fitting the single scattering spectrum using Equations [2405a] and [2405b].
Figure 2405a shows the band gap determination of the HfO_{2} layer in a Si/SiO_{2}/HfO_{2}/polyGe stack by using Equation 2405a (direct band gap [E_{g} = 5.26 eV] for HfO_{2}).
Figure 2405a. Band gap determination of the HfO_{2} layer in a Si/SiO_{2}/HfO_{2}/polyGe stack. Adapted from [1]
[1] Marie C. Cheynet, Simone Pokrant, Frans D. Tichelaar, and JeanLuc Rouvière, Crystal structure and band gap determination of HfO2 thin films, Journal of Applied Physics 101, 054101 (2007).
