Hydrodynamical modeling of charge carrier transport in semiconductors

Enhanced functional integration in modern electron devices requires an accu rate modeling of energy transport in semiconductors in order to describe hi gh-field phenomena such as hot electron propagation, impact ionization and heat generation in the bulk material. The standard drift-diffusion models c annot cope with high-field phenomena because they do not comprise energy as a dynamical variable. Furthermore for many applications in optoelectronics one needs to describe the transient interaction of electromagnetic radiati on with carriers in complex semiconductor materials and since the character istic times are of order of the electron momentum or energy flux relaxation times, some higher moments of the distribution function must be necessaril y involved. Therefore these phenomena cannot be described within the framew ork of the drift-diffusion equations (which are valid only in the quasi-sta tionary limit). Therefore generalizations of the drift-diffusion equations have been sought which would incorporate energy as a dynamical variable and also would not be restricted to quasi-stationary situations. These models are loosely speaking called hydrodynamical models. One of the earliest hydr odynamical models currently used in applications was originally put forward by Blotekjaer [1] and subsequently investigated by Baccarani and Wordeman [2] and by other authors [3]. Eventually other models have also been invest igated, some including also non-parabolic effects [4-6, 8-20]. Most of the implemented hydrodynamical models suffer from serious theoretical drawbacks due to the ad hoc treatment of the closure problem (lacking a physically c onvincing motivation) and the modeling of the production terms (usually ass umed to be of the relaxation type and this, as we shall see, leads to serio us inconsistencies with the Onsager reciprocity relations). In these lectur es we present a general overview of the theory underlying hydrodynamical mo dels. In particular we investigate in depth both the closure problem and th e modeling of the production terms and present a recently introduced approa ch based on the maximum entropy principle (physically set in the framework of extended thermodynamics [21, 22]). The considerations and the results re ported in the paper are exclusively concerned with silicon.