Relaxation time bias field effects

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

Figure 6. Integral (xjn( solid lines) and effective i t v filled circles and asterisks) relaxation times vs. E, for normal rotational diffusion in a dc bias field. Equations (187)—(190) have been used in the calculation.

Here rp(k) is the momentum relaxation time which is due to the electron-phonon and electron-impurity scattering,stands for the electron distribution functions of spin a, h(k) is the DP term which serves as an effective magnetic field and is composed of the Dresselhaus term [10] due to the bulk inversion asymmetry (BIA) and the Rashba term [11] due to the structure inversion asymmetry (SIA),... 

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