User:DarPiv/sandbox

= Urbach Tail = Urbach Tail-is an exponential part in the absorbtion coefficient curve. This tail appears near the optical band edge, in the amorphous, disordered and crystaline materials, because these materials have localized states extended in the band gap.

Urbach Empirical Rule
The width of localized states near the band edges can be expressed by temperature and absorption coefficient :

$$\alpha(h\nu, T) = \alpha_0 exp \left(\sigma \frac{h\nu - E_0}{KT}\right) = \alpha_0 exp \left(\frac{h\nu - E_0}{E_U}\right)$$

In the low photon energy range, the dependence of the absorbtion coefficient ($$\alpha$$) and the photon energy ($$h\nu$$) is given by the following equation:

$$\alpha = \alpha_0 exp\left(\frac{h\nu}{E_U}\right)$$

where $$\alpha_0$$ is a constant, and $$E_U$$ is the Urbach energy (energy of the band tail). This relation is known as Urbach empirical rule.

Urbach energy is weakly dependent upon temperature, and is often interpreted as the width of the band tail due to localized states in the normally band gap that is associated with the disordered or low crystalline materials. Taking the logarithm of the two sides of this equation, one can get a straight line equation:

$$\ln \alpha = \ln \alpha_0 + \frac{h\nu}{E_U}$$

Therefore, the band tail energy or Urbach energy ($$E_U$$) can be obtained from the slope of the straight line of plotting ln (α) against the incident photon energy ($$h\nu$$).

History
Researchers were questioning the nature of the disordered semiconductors' tale states since the nineteen-fifties. It has been found that such tails arise from the strains sufficient to push local states past the band edges. In 1953 F.Urbach identified that such tails decay exponentially into the gap. Later photoemission experiments delivered absorbtion models displaying temperature dependence of the tail. Variety amorphous and imperfect crystalline solids reveal exponential band edges via optical absorption. The surprising universality of this feature suggested the possibility of a common cause. Several attempts to explain the phenomenon were limited, and could not connect specific topological units to the electronic structure.

Recent studies showed that topological filaments (structural patterns that resemble hydrodynamic ﬂow ﬁelds) can be a structure underlying the Urbach edge. However, TF might be not unique in producing the Urbach tail.