Urbach rule

If we assume that P(F) is a delta function centered around F, it follows that a exp [— const. E/F ] this is the energy dependence observed in the Urbach rule. By numerical calculations, Dow and Redfield (1972) showed that this is true also in the relevant part in the spectrum in the case that the electric fields have a Gaussian distribution with reasonable parameters F in this case is /= as in Eq. (4.19). 

Fig. 19.13. Absorption coefficient of EuSe single crystals (K < 10 cm ) and thin films (K > 10 cm" ) at 300 K and 27 K. Verification of the Urbach rule (after Batlogg and Wachter, 1976).

A UV-visible spectrometer is used to measure electronic transition absorption. Electronic transition, UV absorption, loss values for core-composing materials, such as monomer, polymerization initiators, and chain transfer agents, and for polymer rods decrease exponentially with wavelength. A linear relationship is observed for PMMA raw materials. However, PMMA shows an absorption maximum near 360 nm, and linear parts are observed in both lower and higher wavelength regions. This absorption maximum may be derived from residual azo compound. In these cases, a linear part obeys the so-called Urbach rule i.e., electronic transitional loss values,... 

Further complication in semiconductor band shape analysis concerns the spectral region near the fundemental absorption onset. Ideal semiconductor crystal at 0 K should not absorb any photons with energies lower than Eg. Real systems, however, show pronounced absorption tails at energies lower than the bandgap energy (Figure 7.7). The absorption profile within the tail region can be very well approximated by the empirical Urbach s rule [23-26] ... 

Skettrup T. Urbach s rule derived from thermal fluctuations in the band-gap energy. Phys Rev B 1978 18 2622-31. 

Mahan GD. Phonon-broadened optical spectra Urbach s Rule. Phys Rev 1966 145 602-8. 

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