A Hydrodynamical Model for Holes in Silicon Semiconductors: the case of parabolic warped bands Giovanni Mascali (1) * and Vittorio Romano (2) † (1) Dipartimento di Matematica, Universit`a della Calabria and INFN-Gruppo c. Cosenza, 87036 Cosenza, Italy (2) Dipartimento di Matematica e Informatica, Universit`a di Catania, viale A. Doria 6, 95125 Catania, Italy July 16, 2010 Abstract In this paper we present a hydrodynamical model which describes hole motion in silicon. The model is based on the moment method and the closure of the system of moment equations is obtained by using the maximum entropy principle (hereafter MEP). The heavy, light and split-off valence bands are considered. The first two are described by taking into account their warped shape, while for the split-off band a parabolic approximation is used. Moreover the model is coupled with an analogous model for electrons, so obtaining a complete description of charge transport in silicon. Numerical simulations are performed both for bulk silicon and a p-n junction. Key Words: hole transport; hydrodynamical models: diode. PACS: 72.20 1 Introduction While there is a very wide literature on electron transport in semiconductors, hole transport has been studied less intensively, in particular there is a lack of macroscopic models. This is probably due to the complex structure of the valence bands which makes the study rather difficult. However holes give a relevant contribution to the charge transport properties in a great variety of semiconductor devices: silicon and compound p-channel field-effect transistors, bipolar transistors, hetero-structural bipolar transistors, and optoelectronic devices like lasers and light emitting diodes. A powerful tool to describe the behavior of those devices consists of the system of the semi-classical Boltzmann transport equations for electrons and holes coupled to the Poisson equation. However, al- though modern computers operate at continuously increasing CPU speed, the direct integration of this system is a very expensive computational task. For this reason, it is appropriate to develop macroscopic models, see for example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and references therein. In industrial ap- plications for simulating the hole transport in bipolar devices the drift-diffusion model is commonly used [14, 15], which assumes that the charge flow is isothermal. This is justified in devices like n-MOSFET’s (Metal Oxide Field Effect Transistors), where the hole contribution to the total current is marginal. However, as said, in other devices, like bipolar and p-MOSFET’s, the role of holes in charge transport is of the same order or even greater than that of electrons. In such situations more sophisticated models are needed. In this paper we revisit the hydrodynamical model of hole transport in silicon semiconductors pre- sented in [16], in the sense that we take a correct expression for the acoustic phonon energy (see the * e-mail: g.mascali@unical.it † e-mail: romano@dmi.unict.it 1