Maximum-entropy principle for static and dynamic high-field transport in semiconductors

By introducing generalized kinetic fields we have developed an extended HD approach based on the maximum entropy principle, which includes an arbitrary number of moments of the distribution function. Then, for the perturbation of these moments a set of coupled balance equations is constructed and analytical expressions for all the small signal coefficients are obtained in the time and frequency domains. In particular, the generalized expressions for the response matrix , the perturbing forces −e E t E, the response functions Kt, and the differential mobilities =X +iY are directly calculated for any moment of interest. By generalizing previous results, the theory provides also some relations in integral form and in asymptotic form that are used to describe the small signal analysis. From the knowledge of the quantities K0, X0 and of the derivatives dK / dt0+, d2K / dt20+, dY /d 0, d2X /d 20 the anatomy of all response functions and differentials mobilities is inferred in the time and frequency domains. The power of the present approach stems from the construction of an algebraic in place of an integral formulation of the theory. Thus, from the explicit knowledge of , E the small signal coefficients are consistently obtained in algebraic form. The theory is formulated at a kinetic level, without the need to introduce external parameters, and it has been carried out within a total energy scheme, thus using an energy dispersion of general form full-band approach. The physical plausibility of the theory has been confirmed by analysing the high field transport in n-Si. To this purpose, as generalized kinetic fields we have taken the independent quantities =pui1¯uis and as unique independent mean quantities the traceless moments F =Fpi1¯is, where the indices p and s are associated with the isotropic and deviatoric parts of the tensors, respectively. The analysis of dc and ac numerical results shows that the behavior of all moments is determined essentially by the competition between the action of the electric field and that of the dissipative scattering processes. In particular, the action of the electric field prevails on the moments which have an increasing isotropic part while the action of dissipative processes are more evident in the moments with a large deviatoric part. By studying the eigenvalues and eigenvectors of the response matrix we have analyzed the coupling among the different macrovariables moments and we have found that this coupling leads to a nonexponential decay of the corresponding response functions. In particular, by considering the moments F0s, with null isotropic part and increasing deviatoric part, the combined action of the electric field and dissipative processes has been quantitatively investigated. We have thus demonstrated that, at high fields: i the vanishing dc differential mobility of different moments, ii the presence of complex eigenvalues, iii the negative values taken by the response functions, iv the positive overshoot of differentials responses, and v the maximum of the real and imaginary parts of the ac differentials mobility, are all related to the efficiency of dissipative scattering processes. Analogously, by considering the moments Fps with increasing isotropic part p1 we have established that a simple relaxation approach based on a single time scale loses of validity, because both the response functions and the corresponding differential responses evolve with different time scales. In particular, the energy response function KW ˜ evidences the streaming character of hot carriers through a nonmonotonic behavior with a maximum which separates different time scales. The limits of the concept of a single relaxation time are also evident in the shape of the corresponding ac differential mobilities which show a nonregular behavior at increasing frequencies before reaching the cutoff. The theory has been validated by comparing the present results with those obtained from MC simulations and with available experiments for the standard quantities of direct physical interpretation v,W ˜ ,S˜. Therefore, we believe that the present approach represents a useful standard to obtain a generalized modeling of the relevant static macrovariables of interest and of the small-signal dynamics coefficients in terms of rigorous analytical formulas associated with microscopic peculiarities of the single carrier transport. In addition to offering an approach complementary to existing kinetic method based on Monte Carlo simulations and/or iterative solutions of the Boltzmann equation, the theory has the advantages of providing a systematic framework to investigate transport phenomena under far from equilibrium conditions and of operating within a contained computational environment.