Determination of Band Gap from Low-loss Spectra in EELS
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Several methods have been applied to determine the band gap from low-loss spectra in EELS technique.

For some materials (especially wide band gap materials), the band gap (Eg) from low-loss spectra obtained by EELS technique can be determined by eye observation of the SSD (single scattering distribution) spectrum. In this case, Eg 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.

Schematic illustration of Si and Ge SSD spectra in the low-energy range

Figure 2406a. Schematic illustration of Si and Ge SSD spectra in the low-energy 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,
        Schematic illustration of Si and Ge SSD spectra in the low-energy range for a direct band gap, ------------- [2406a]
and,
        Schematic illustration of Si and Ge SSD spectra in the low-energy range for an indirect band gap ------------- [2406b]
where,
        I0 and c -- The constants.
        E -- The energy loss.

Therefore, Eg can be extracted by fitting the single scattering spectrum using Equations [2406a] and [2406b]. Figure 2406b shows the band gap determination of the HfO2 layer in a Si/SiO2/HfO2/poly-Ge stack by using Equation 2406a (direct band gap [Eg = 5.26 eV] for HfO2).

Schematic illustration of Si and Ge SSD spectra in the low-energy range

Figure 2406b. Band gap determination of the HfO2 layer in a Si/SiO2/HfO2/poly-Ge 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 zero-loss peaks (ZLPs) can create large, random data spikes in the deconvoluted spectrum.

It is very general that the band gap is determined based on Kramers-Kronig analysis. For instance, Figure 2406c shows the EEL spectra of plasmon region (zero-peak 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 Fourier-log deconvolution method.

EEL spectra of plasmon region for (a) crystalline and (b) amorphous diamond
EEL spectra of plasmon region for (a) crystalline and (b) amorphous diamond
(a)
(b)

Figure 2406c. 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, ε2, associated with a single electron excitation of an interband transition, while EELS cannot directly give ε2. Based on the obtained loss function, the real part1), and the imaginary part (ε2) of the dielectric function for the crystalline and the amorphous diamond are extracted using Kramers-Kronig analysis as shown in Figure 2406d. Table 2406 lists the onset energy (indicating the band gap energy) and the peaks in the imaginary spectrum obtained from Figure 2406d and the original interband transitions.

imaginary part (ε2) of the dielectric function for (a) crystalline and (b) amorphous diamond
imaginary part (ε2) of the dielectric function for (a) crystalline and (b) amorphous diamond
(a)
(b)

Figure 2406d. Real part1), and the imaginary part (ε2) of the dielectric function for (a) crystalline and (b) amorphous diamond.

Table 2406. Onset energies (band gap energies), peaks in the imaginary spectrum in Figure 2406d 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

 


 

[1] Marie C. Cheynet, Simone Pokrant, Frans D. Tichelaar, and Jean-Luc Rouvière, Crystal structure and band gap determination of HfO2 thin films, Journal of Applied Physics 101, 054101 (2007).
[2] U. Bangert and R. Barnes. Electron energy loss spectroscopy of defects in diamond. Phys. Stat. Sol. (A), 204 (7): 2201–2210, 2007.

 

 

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