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Semiconductors

Recall the equatio for absorbing materials. Just by assuming oscillating charges in the material we found that the dielectic constant actually changes with frequency. Also here a more realistic model would assume that tha dielectric function differs from one at high energies by replacing it with ε₀₀.

The figure illustrates general behaviour of the dielectric function and the refractive index.

The figure below shows the dielectric properties of silicon between 1 and 5 eV.

By comparing the two figures we find that the description of the simple model is adequate between 1 and 3 eV, but at higher energies it can merely illustrate the general trends rather than account for the observed features. A somewhat better description of silicon can be obtained by assuming two or three oscillators with different strengths.

Let us look more closely at the absorption coefficient. The figure below compares the simple model and the actual data of silicon.

In the logarithmic plot we can clearly see the difficulties of the simple model because silicon does not absorb below the band gap energy of 1.14 eV. Above the gap the absorption coefficinet increases and eventually agreement with the behaviour of the oscillator model is seen.

Absorption in semiconductors

Obviously, absorption in semiconductors is more complicated. Also, we must distinguish direct and indirect absorption processes. So far we only considered the energy which is absorbed, but we must also keep in mind the conservation of the momentum. Compared to electrons in the crystal the momentum of photons is negligible, in the band diagram this is essentially a vertical contribution only. Thus, a direct process is possible whenever the photon energy is sufficient to excite an electron into a free state of almost same k-value, which is simply straight up.

The theory gives a sqare-root type of relation between absorption coefficient and photon energy, offset by the bandgap.

In indirect semiconductors the conduction band maximum and the valence band minimum are not located at the same position in k-space. Photons with sufficient energy to excite the electron must still fulfill the conservation of momentum which is possible by the additional absorption or emission of a phonon.

In this case the theory results in a second order equation and also takes into account the energy of the phonon.

Note: It would be more accurate to consider absorption and emission of a phonon. This would require to add the two contributions after multiplying with the phonon probabiltiy distribution, giving an additional dependence on temperature. Absorption studies of indirect semiconductors like germanium show two distinct absorption regions and allow to evaluate the energy of the involved phonon.

The figure below compares the absorption coefficient a of different semiconductors. The blue curve represents silicon which is an indirect semiconductor. The dashed blue line has been calculated from the simple oscillator model in the paragraph above and obviously does not include the reduced absorption of silicon around the bandgap (observed in logharithmic scale in the right panel). The red curve corresponds to a direct semiconductor like CuInSe₂ and exhibits much higher absorption.

A better description of the dielectric properties and absorption phenomena includes the particulars of the band structure like the different types of critical points in general [1] as well as the various rules for transitions between such points (see [2] for an example of CuInSe₂ and CuGaSe₂) and also excitonic states of electron hole pairs (see e.g. [3] for a treatment of ZnS).

[1] S. Loughin, R. H. French, L. K. De Noyer, W.-Y. Ching, and Y.-N. Xu, J. Phys. D: Appl. Phys. 29 (1996) 1740

[2] T. Kawashima, S. Adachi, H. Miyake, and K. Sugiyama, J. Appl. Phys. 84(9) (1998) 5202

[3] T. Tsuchiya, S. Ozaki, and S. Adachi, J. Phys.: Condens. Matter 15 (2003) 3717

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