On the Generalized Distribution Functions in Heavily Doped Nano Materials at Terahertz Frequency

Abstract

In this chapter, we present a simplified analysis of the generalized distribution function of the carriers in heavily doped materials in the form $F(\overline{E}) = [1 + A + \exp (y)]^{ - 1}$ where $A$ is a constant and the other variables are defined in the text. The substitution $A = 0$ and $- 1$ lead to the well-known Fermi–Dirac statistics and Maxwell–Boltzmann distribution, respectively. The substitutions $A = 0$ and $y = 0$ together with $A = - 4$ and $y = 0$ lead to the well-known Pauli’s exclusion principle ($\pm (1/2)$), whereas the substitutions $A = i - 1$ $(i = \sqrt { - 1} )$ and $y = 0$ together with $A = - 3 - i$ and $y = 0$ lead to the complex values of the Pauli’s spin in the tail zone as $\pm (1 - i)/2$, respectively. Because of the complex Pauli’s spin value, the electron energy component due to the spin $g$ factor along the direction of the magnetic field $B$ in heavily doped electronic materials forming band tails in the tail zone generates the magnitude of the electron energy as 71% g times the cyclotron resonance energy together with the phase value $\pm \,(\,\pi /4)$ in this case. We have also shown the Bose Einstein statistics in this context. Besides, the cases of terahertz frequency, heavy doping and intense electric field can be covered by replacing the value of $\overline{E}$ under the mentioned conditions.