Electronic and Optical Interband Transitions and Measured by EELS
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Interband transitions are excited from occupied valence bands and shallow core levels to the unoccupied conduction bands. The electronic and optical interband transitions are categorized by direct and indirect transitions because the conduction band bottom and the valance band top contain more than one extrema. The electronic and optical band gaps can be calculated by the predominant mechanism of band to band transitions and are normally less than several volts. In the allowed direct transition, the electrons in the valence band transit vertically to the conduction band under the excitation of incident electrons or photons. The absorption coefficient as a function of photon energy in the allowed direct transition can be given by [1],

       electronic and optical interband transitions ------------------------------ [3952a]

where,
       A -- A constant
       v -- The frequency
       h -- The Planck constant
       Egd -- The allowed direct energy gap.

The interband transitions correspond to the absorption peaks in the imaginary part of dielectric function ε2. On the other hand, it is strongly modulated by ε21 + ε22 in energy loss function, which dumps the peak strength of interband transitions.

In the indirect transition, the electrons transit from the valence band top to the conduction band bottom with participation of photons and appropriate phonons. In this case, the absorption coefficient can be given by [2],

       electronic and optical interband transitions ------------------------------ [3952b]

where,
       B -- A constant
       Egi -- The indirect energy gap
       Ep -- The energy of the absorbed (+) or emitted (-) phonons.

Interband transitions are an effect of the lattice. They shift the plasmon energy and can be included via the Drude–Lorentz theory. [3 - 5] The energy region of the EEL spectrum (EELS) up to the energy loss of ∼50 eV is dominated by the collective excitations of valence electrons (plasmon) and by interband transitions. In order to include interband transitions, we need to have a good understanding of the electronic structure. The EELS of the low-energy loss region less than 50 eV is particularly called valence electron energy loss spectroscopy (VEELS).

Table 3952. Energy losses and scattering angles of various inelastic electron scatterings in electron interaction with materials.
Process Phonon excitation Inter of intra band transitions Plasmon excitation Inner shell ionization
Region of energy loss
Low loss (< 50 eV)
High loss (> 50 eV)
Energy loss E [eV] ~0.02 3 - 25 5 - 25 10 - 2000
Scattering angle θE [mrad] 5 - 15 5 - 10 < 0.1 0.1 - 10
Oscillation Collective oscillations of atoms (e.g. lattice vibrations)   Collective oscillations of free electrons, a quantum of a collective longitudinal
wave in the electron gas of a solid
 
Detectability by EELS Is not resolved Yes Yes Yes
Effects Causes specimen to heat up      
Time     Damped out in < 10−15 s  
Localization     Localized to < 10 nm  
Interaction     Most common inelastic interaction due to high free electron density  
Cross sections     Relatively large Relatively small
Mean-free
paths
    Relatively short Relatively large
Intensity     Much intense Much smaller
Materials     Predominant in metals  
Modification     Can reduce the number of phonons by cooling the specimen.  
Other names Thermal diffuse scattering      
Characteristics Diffuse background, don’t carry any useful information Signature of the structure   Elemental information

Figure 3952a 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 3952a. 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 3952b. Table 3952 lists the onset energy (indicating the band gap energy) and the peaks in the imaginary spectrum obtained from Figure 3952b 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 3952b. Real part1), and the imaginary part (ε2) of the dielectric function for (a) crystalline and (b) amorphous diamond.

Table 3952. Onset energies (band gap energies), peaks in the imaginary spectrum in Figure 3952b 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

Note if the interband transitions are in the energy range less than ∼ 2 eV, these spectral features are very close to the intense zero-loss peak from elastic scattering, so that it is difficult to resolve them.

 

 

 

 

 

 

 

[1] Tauc JC (1972) Optical properties of solids. North-Holland, Amsterdam.
[2] El-Korashy A, El-Fadl AA (1999) Phys B 271:205.
[3] R.F. Egerton, Electron Energy Loss Spectroscopy in the Electron Microscope, Plenum Press, New York, 1996.
[4] P. Schattschneider, B. Jouffrey, in: L. Reimer (Ed.), Energy-filtering Transmission Electron Microscopy, Springer, 1995, p. 151.
[5] H. Raether, Excitation of Plasmons and Interband Transitions by Electrons, Springer-Verlag, Berlin, 1980.

 

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