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Urbach Tail

Band-gap model (blue dotted line), the Urbach-tail extension (red dotted line), and the band-gap model with Urbach tail (black solid line).

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.