Maximum entropy principle within a total energy scheme for hot-carrier transport in semiconductor devices

By means of generalized kinetic fields {/spl epsi//sup p/, /spl epsi//sup p/u(i/sub 1/),..., /spl epsi//sup p/u(i/sub 1/) u(i/sub 2/)...u(i/sub s/)...} (where u/sub i/ is the carrier group velocity, /spl epsi/(k/spl I.oarr/) the single particle band energy, p=0,1,...N, and s=1,2,...M with arbitrary values of N and M) we present a general formulation of the maximum entropy principle. A new system of generalized hydrodynamic (HD) equations is derived with the full complexity of the band modeled in terms of a single particle with an effective mass which is a function of the average total energy and becomes a new constitutive function. Present HD theory thus does not need other adjustable parameters but the knowledge of the elementary microscopic interactions as for kinetic theory. In the bulk material stationary and small signal kinetic coefficients are consistently obtained as a function of the external electric field. In the context of the small signal analysis we have introduced a generalized response matrix and evaluated numerically the linear response functions of the different moments in the time domain. The validity of this approach has been confirmed by the satisfactory agreement with the numerical results of full-band Monte Carlo (MC) simulations and available experimental data for the case of electrons in Si (bulk and n/sup +/nn/sup +/ structures) at T/sub 0/=300 K. As first step, we consider the first 13 moments of the distribution function and we report the velocity /spl upsi/, the energy W/spl tilde/ and the energy flux S/spl tilde/ for three n/sup +/nn/sup +/ Si structures with channel lengths of 0.2, 0.3 and 0.4 /spl mu/m and an applied voltage of 1.5 V and 2 V, respectively. For the HD calculations both parabolic and nonparabolic band models are reported. For the longer structure MC results of the analytical nonparabolic model with the same scattering parameters used in the HD calculations are also reported. When compared with MC results, HD calculations are found to agree satisfactorily apart from some discrepancy near the anode region. These discrepancies are associated with the strong gradient of the electric fields which would require to account for other moments in the development of the analytical approach. As second step, we consider the contribution of higher moments and we report a relevant subset of a complete series of data showing the spatial profiles of the energy flux S/spl tilde/ for the n/sup +/nn/sup +/ structure calculated for an increasing number of moments with M=1, N=1 and N=5 at increasing applied voltages. Overall, the agreement between the HD and MC results is considered to be satisfactory, thus validating the constitutive relations found here.