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For some materials (especially wide band gap materials), the band gap (E_{g}) from lowloss spectra obtained by EELS technique can be determined by eye observation of the SSD (single scattering distribution) spectrum. In this case, E_{g} is defined as the energy corresponding to the first onset in the spectrum, for instance, 0.67 and 1.12 eV for Ge and Si respectively, in the schematic spectra in Figure 2406a.
Figure 2406a. Schematic illustration of Si and Ge SSD spectra in the lowenergy range.
On the other hand, the band gap can also be determined based on Bethe’s theory. In this case, the SSD spectrum is described to be proportional to the product of the joint density of states (JDOS). Here, the JDOS is given by (e.g. Figure 2406b),
for a direct band gap,  [2406a]
and,
for an indirect band gap  [2406b]
where,
I_{0} and c  The constants.
E  The energy loss.
Figure 2406b. EELS intensity recorded from cubic GaN by using a monochromated TEMEELS system, presenting direct and indirect electronic transitions. [14] 
Therefore, E_{g} can be extracted by fitting the single scattering spectrum using Equations [2406a] and [2406b].
Figure 2406c shows the band gap determination of the HfO_{2} layer in a Si/SiO_{2}/HfO_{2}/polyGe stack by using Equation 2406a (direct band gap [E_{g} = 5.26 eV] for HfO_{2}).
Figure 2406c. Band gap determination of the HfO_{2} layer in a Si/SiO_{2}/HfO_{2}/polyGe stack. Adapted from [1] 
However, at very low energy losses, especially in the range of 0 to 5 eV, the current deconvolution methods for multiple inelastic scattering are not sufficiently accurate [2]. For instance, in this energy range, any differences between the experimental and modelled zeroloss peaks (ZLPs) can create large, random data spikes in the deconvoluted spectrum.
As shown above, several methods have been applied to determine the band gap from lowloss spectra in EELS technique. [37] However, such technique is not widely used for band gap measurements because of the presence of “parasitic” unwanted losses which are superimposed to the band gap signal. Fortunately, according to the simulations based on the socalled Kröger formula [17], all the unwanted losses, with only a small momentum transfer, happen at very low scattering angles and can, in general, be avoided by not using the scattering events in that small scattering range (in ~ μrad range) [11, 18]. That is, there is a large difference in scattering angles between the interband and “parasitic” signals so that the evaluation of band gap can be improved with a carefully chosen angular range to collect the electrons scattered away from the forward direction. [15, 16]
Table 2406a. Suggested solution of avoiding unwanted energy losses (in a ~ μrad semiangle range) in band gap determination with lowloss spectra in EELS technique. 
Factors or artifacts affecting measurement accuracy 
Solution 
Reference 
Unwanted losses through the emission of Cherenkov radiation or the excitation of surfaceguided light modes 
Minimized by using an annular or offaxis
collection aperture 
[11] 
Minimized by subtracting spectra recorded with small and large collection
apertures 
[11] 
Subtracted spectra recorded
with two different collection angles 
[13] 
Zeroloss peak 


Zeroloss peak tails extended to several eV 
Minimized with a monochromator 
[10, 12] 
Minimized by setting the parallelrecording spectrometer to a high enough energy dispersion (low
eV/channel) 
[10] 
Remove the highenergy tail by fitting and subtraction or by Fourier ratio deconvolution 
[11] 
Band gap (E_{g}) position 
Obtained with the intersection point of linear fits to the intensity rise and
background, e.g. Figure 2406d. 
[12] 
Band gap mapping limited by delocalization of inelastic scattering 


Figure 2406d. Bandgap
measurements on SiO_{2} (silicon
oxide) films of different
thicknesses. The band gap
(8.9 eV) was obtained with the intersection of linear fits to the intensity rise and
background. [12] 
It is very general that the band gap is determined based on KramersKronig analysis. For instance, Figure 2406e shows the EEL spectra of plasmon region (zeropeak is not shown) for crystalline and amorphous diamond. The loss function (Im[l/ε]) can then be obtained by removing contribution from multiple scattering using the Fourierlog deconvolution method.
Figure 2406e. EEL spectra of plasmon region for (a) crystalline and (b) amorphous diamond.
An optical absorption method can directly provide the imaginary part of the dielectric function, ε_{2}, associated with a single electron excitation of an interband transition, while EELS cannot directly give ε_{2}. Based on the obtained loss function, the real part (ε_{1}), and the imaginary part (ε_{2}) of the dielectric function for the crystalline and the amorphous diamond are extracted using KramersKronig analysis as shown in Figure 2406f. Table 2406b lists the onset energy (indicating the band gap energy) and the peaks in the imaginary spectrum obtained from Figure 2406f and the original interband transitions.
Figure 2406f. Real part (ε_{1}), and the imaginary part (ε_{2}) of the dielectric function for (a) crystalline and (b) amorphous diamond. 
Table 2406b. Onset energies (band gap energies), peaks in the imaginary spectrum in Figure 2406f and the original interband transitions. 

Onset energy (band gap energy) 
Peaks in imaginary spectrum 
Interband transition 
Crystal diamond 
5.5 eV 
8.2 
Γ point 
12.7 
X and L points 
Amorphous diamond 
4.0 eV 
7.2 eV 
Γ point 
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