By Walter Allegretto, Christian Ringhofer, Angelo Marcello Anile, Angelo Marcello Anile
The C.I.M.E. consultation on Mathematical difficulties in Semiconductor Physics, held in Cetraro (Italy) July 15-22, 1998 addressed researchers with a powerful curiosity within the mathematical features of the idea of service delivery in semiconductor units. The lined topics comprise hydrodynamical types for semiconductors in line with the utmost entropy precept of prolonged thermodynamics, mathematical idea of drift-diffusion equations with functions, and the equipment of asymptotic research.
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Extra resources for Mathematical problems in semiconductor physics: lectures given at the C.I.M.E. summer school held in Cetraro, Italy, July 15-22, 1998
9. A. Majorana, Space homogeneous solutions of the Boltzmann equation describing electron-phonon interactions in semiconductors, Transp. Theory Stat. Phys. 20 (1991) pp. 261-279. 10. A. Majorana, Conservation laws from the Boltzmann equation describing electron-phonon interactions in semiconductors, Transp. Theory Stat. Phys. 22 (1993) pp. 849-859. 11. A. Majorana, Equilibrium solutions of the non-linear Boltzmann equation for an electron gas in a semiconductors, Il Nuovo Cimento 108B (1993) pp.
By introducing a MUSCL interpolation, we approximate v(x, t) by a piecewise linear polynomial Lj (x, t) = vj (t) + (x − xj ) 1 v , ∆x j xj− 12 ≤ x ≤ xj+ 12 . (84) 1 (xj + xj+1 ) 2 (85) with xj− 12 = 1 (xj−1 + xj ) 2 xj+ 12 = and in order to ensure a second order accuracy we require that 1 ∂ v = v(xj , t) + O(∆x). ∆x j ∂x (86) Therefore, Eq. (81), together with (82), (83), and (84), gives 1 1 [vj (t) + vj+1 (t)] + v − vj+1 + 2 8 j 1 1 −λ f (vj+1 (tn ) − λfj+1 ) − f (vj (tn ) − λfj ) + O(∆t3 ). 2 2 v j+ 12 (t + ∆t) = Because the initial state at t = tn is given by the piecewise linear function Lj (x, tn ), the ﬂuxes remain regular functions if the solutions of the corresponding generalized Riemann problems between adjacent cells do not interact.
5 4 Fig. 18. The time evolution of the electron average velocity for diﬀerent values of the electric ﬁeld and for N+ = 1014 /cm3 and 1017 /cm3 respectively. The stationary regime is reached in a few picoseconds. The typical phenomena of overshoot and saturation of the velocity are both qualitatively and quantitatively well described. We also report the curves representing the electron valley occupancy, nΓ 4nL nΓ +4nL and nΓ +4nL , and average velocity as functions of the electric ﬁeld (see Figs.