In amorphous semiconductors and insulators, the using conductivity formulas are semi-empirical and have no satisfying physical explanations. A conductivity equation has been derived by Debye for the response of ideal materials which is rarely observed in practice, but a general conductivity equation which includes the previous empirical equations via a correct choice of arbitrary parameters and moreover totally theoretical derivation had to be generated. Hence, to determine the motion of electrons in the amorphous environment, we defined the equation of motion including viscous forces as a function of coordinates, their derivatives and time variables. We developed a fractional form of this equation over these three variables and finally obtained the most generalized equation of motion, which counts the overall interactions by a fractional form as a variation of two variable. The improved formula, called the stretched Havriliak–Negami equation, has the same form and behavior as the semi-empirical equation and reducible to the Cole–Cole and Cole–Davidson-type of conductivity.
Thalium selenide (TlSe), which has a lattice with tetragonal symmetry, is a member of the A3B6 semiconductor group. The structure of TlSe is defined as chains where atoms inside are bonded with an ionic-covalent bond. TlSe thin films were deposited by thermal evaporation under a high vacuum on glass substrates. The structure of TlSe thin films is amorphous with a tetragonal structure. The AC conductivity measurements were operated via the measurements of capacitance and dielectric dissipation (tanδ) at room temperature. AC conductivity values change between 10−11 and 10−6 S/cm at the low-frequency side with decreasing thickness. Two different conduction regions were observed with increasing frequency. The region observed at the low-frequency side can be attributed to the motion of a chain-like part of the lattice, while the region observed at the high-frequency side can be attributed to side groups.
H-index has become more popular nowadays and is used for some scientific performance criteria in the world widely. This indexing method does not correctly measure any performance or carrier specifications because of the parameters that are used to form the measurement basis. H-index is located based on citation(C) and paper(N) parameters that involve no logical criterion on the counting process and so measurement on this basis can only give quantity results not any quality information. Therefore, we need a new indexing instrument to find out also the scientific quality unique to an individual author even if that takes into account the effect of multiple coauthorships. Ipso facto, we create a new bibliometric indicator or academic performance indicator called the u-index.
We aim to derive a phenomenological approach to link the theories of anomalous transport governed by fractional calculus and stochastic theory with the conductivity behavior governed by the semi-empirical conductivity formalism involving Debye, Cole-Cole, Cole-Davidson, and Havriliak-Negami type conductivity equations. We want to determine the anomalous transport processes in the amorphous semiconductors and insulators by developing a theoretical approach over some mathematical instruments and methods. In this paper, we obtain an analytical expression for the average behavior of conductivity in complex or disordered media via using the fractional-stochastic differential equation, the Fourier-Laplace transform, some natural boundary-initial conditions, and familiar physical relations. We start with the stochastic equation of motion called the Langevin equation, develop its equivalent master equation called Klein-Kramers or Fokker-Planck equation, and consider the time-fractional generalization of the master equation. Once we derive the fractional master equation, then determine the expressions for the mean value of the variables or observables through some calculations and conditions. Finally, we use these expressions in the current density relation to obtain the average conductivity behavior.
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