The band tails

The existence of localized states was predicted early on in the studies of amorphous semiconductors by the Anderson localization theory (Section 1.2.5) and their presence is well established ex- 

The probability of electron emission into the vacuum jumps abruptly at the work function energy, vao J l d is approximately constant above this energy, so that the valence band density of states is given by dT(A D)/dA . 

An example of the valence band tail of a-Si H obtained using this technique is shown in Fig. 3.9. The position of the mobility edge, E, is not obtained in this experiment, which does not distinguish localized from extended states, but is estimated to be at about 5.6 eV. There is a linear density of states near and above y, and an exponential band tail over several orders of magnitude of N E) below y. The slope of the 

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In amorphous semiconductors, information about the width of the band tail states (or disorder) may also be extracted from the optical absorption spectra. For photon energies near bandgap energy, the optical absorption coefficient of amorphous semiconductors exhibit an exponential dependence on the photon energy, following the so-called Urbach relationship ... 

Economou et al. (1985) and Soukoulis et al. (1985, 1986, 1987) have used somewhat similar methods to calculate both the density of states, the mobility edge and the conductivity as a function of energy for the case of diagonal disorder their work is limited to disorder parameters V0 less than one-fifth of the bandwidth B, and is therefore relevant to the band tail... 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I > Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

Fig. 1.6. Schematic density of states distribution for an amorphous semiconductor showing the bands, the band tails, and the defect states in the band gap. The dashed curves are the equivalent density of states in a crystal.

So far no amorphous semiconductors have been made with a Fermi energy in the extended states beyond the mobility edge. The Fermi energy of doped a-Si H moves into the band tails, but is never closer than about 0.1 eV from the mobility edge. There is no metallic conduction, but instead there are several other possible conduction mechanisms, which are illustrated in Fig. 1.11. 

Although carriers cannot conduct in localized states at zero temperature, conduction by hopping from site to site is possible at elevated temperatures. Hopping conduction in the band tail is given by. 

Optical transitions between the valence and conduction bands are responsible for the main absorption band and are the primary measure of the band gap energy. The optical data are also used to extract information about the band tail density of states. However, the absorption coefficient depends on both conduction and valence band densities of states and the transition matrix elements and these cannot be separated by optical absorption measurements alone. The independent measurements of the conduction and valence state distributions described in Section 3.1.1 make it possible to extract the matrix elements and to explore the relation between N E) and the optical spectrum. 

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).

The dark ESR spectra of doped a-Si H in Fig. 5.10 show resonances near g = 2, with different line shapes and g-values from those of the dangling bond (Stuke 1977). These lines are attributed to band tail states because they are observed when the Fermi energy is moved up to the band tails by doping and also in the low temperature LESR spectra of undoped a-Si H, when electrons or holes are optically excited into the band tails. The larger g-shift for the valence band tail states than for the conduction band states is expected from Eq. (4.12). 

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