The Urbach edge

The optical absorption has an exponential energy dependence in the vicinity of the band gap energy. 

Redfield 1970). This model is able to account for the exponential slope with internal fields which are reasonably consistent with the disorder of the amorphous semiconductors. 

Since the slope, E, of the Urbach absorption reflects the shape of the valence band tails, it follows that varies with the structural disorder. For example, one measure of the disorder is the average bond angle variation, which is measured from the width of the vibrational spectrum using Raman spectroscopy (Lannin 1984). Fig. 3.22 shows an increasing E with bonding disorder, which is caused by changes in the deposition conditions and composition (Bustarret, Vaillant and Hepp 1988 also see Fig. 3.20). The defect density is another measure of the disorder and also increases with the band tail slope (Fig. 3.22). A detailed theory for the dependence of defect density on is given in Section 6.2.4. 

---

Fig. 3.23. Temperature dependence of (a) the slope, of the Urbach edge, and (6) the band gap energy and (c) the correlation between the band gap and the band tail slope (Cody et al. 1981).

Eq. (3.35) relates the broadening of the Urbach edge and the shift of the band gap, both of which originate from the thermal and bonding disorder. 

The Urbach edge represents the joint density of states, but is dominated by the slope of the valence band, which has the wider band tail. Expression (3.37) for is therefore also an approximate description of the thermal broadening of the valence band tail. It is worth noting that the slope is quite strongly temperature-dependent above 200 K. This may have a significant impact on the analysis of dispersive hole transport, in which the temperature dependence of the slope is generally ignored. 

The electronic transitions which are responsible for the Urbach edge have been investigated by Izumitani and Hirota (1985) and Izumitani et al. (1986) by UV reflection spectroscopy with synchrotron radiation in Zrp4 BaFj and AlFj-CaFj-P2O5 glasses. In the FZ glasses, in addition to weaker features at 7.3, 9.3, 13.2 and... 

The theories of the Urbach edge are based on the idea that a sharp absorption edge is broadened by some mechanism. In ionic crystals there is little doubt that optical phonons are responsible for the Urbach edges. If their frequency is then by a general argument given below... 

According to the theory of the Urbach absorption edge in crystals, the slope E is proportional to the thermal displacement of atoms r(7). The frozen phonon model assumes that an amorphous semiconductor has an additional temperature independent term, r , representing the displacements which originate from the static disorder, so that... 

Fig. 5.18 compares the optical absorption spectra of phosphorus doped a-Si H with the corresponding compensated material. The compensated a-Si H has a substantially broader Urbach edge, but a... 

Ee is approximately equal to Eo and F is a constant having a range of 10-25 eV . Absorption edges which obey the above exponential equation are called Urbach edges. There is as such no clear understanding of the origin... 

The analysis of the shape of the absorption edge of the high-pressure phase (Fig. 13) shows the existence of two spectral ranges with different types of energy dependence on the absorption coefficient. At high values of absorption it follows the empirical Tauc relation [57] in the case of parabolic band edges (Fig. 13(b)), while at smaller absorption a so-called Urbach or exponential absorption tail [58, 59] is observed (Fig. 13(c)). The existence of this kind of absorption edge is normally related to amorphous semiconductors. The optical absorption gap determined from our experiment is 0.6-0.7 eV and it decreases with pressure (see below). The slope of the Urbach tail, which can be considered as a measure of a random microfield [59] is found to be T=2.6 eV at 160 GPa. This is very close to what one would expect for an amorphous phase with a coordination of 2.5 [59]. 

Fig. 10, Hole photocurrenl transients measured with 2- and 8-V applied bias (solid and open circles, respectively) at 299 K in the time-of-flight experiment. [Reprinted with permission from Solid State Communications, Vol. 47, T. Tiedje, B. Abeles, and J. M. Cebulka, Urbach edge and the density of states of hydrogenated amorphous silicon. Copyright 1983, Pergamon Press, Ltd.]...

The frequency and temperature dependence of Urbach edges is empirically described by the equation... 

---