Urbach's rule derived from thermal fluctuations in the band-gap energy

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Abstract
The exponential absorption edge (known as Urbach's rule) observed in most materials is interpreted in terms of thermal fluctuations in the band-gap energy. The main contribution to the temperature shift of the band-gap energy is due to the temperature-dependent self-energies of the electrons and holes interacting with the phonons. Since the phonon number is fluctuating in thermal equilibrium, the band-gap energy is also fluctuating resulting in an exponential absorption tail below the average band-gap energy. These simple considerations are applied to derive Urbach's rule at high temperatures, while a simplified model with independent, noninteracting atoms is proposed to explain the behavior of Urbach's rule in the whole temperature range. The three parameters entering Urbach's rule are expressed in terms of parameters derived from the temperature shift of the band gap and from the exciton absorption. Comparison with experiments is performed for the II-VI compound ZnO. It is shown that a good agreement is found between the temperature shift of the exciton line observed experimentally and the temperature shift computed from the steepness parameter of Urbach's rule. The agreement with experimental values for the two other parameters is also satisfactory. It is shown that the band-gap shift (and absorption tail) in ZnO is caused by interaction with both acoustical and optical phonons. While the temperature-dependent polaron contribution can account for the optical-phonon contribution, the deformation-potential interaction with LA phonons is not sufficient to account for the acoustical-phonon contribution.
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