DOI 10.1393/ncc/i2010-10578-0 Colloquia: ICTT2009 IL NUOVO CIMENTO Vol. 33 C, N. 1 Gennaio-Febbraio 2010 Hydrodynamic subband model for semiconductors based on the maximum entropy principle G. Mascali( 1 )( 2 ) and V. Romano( 3 ) ( 1 ) Dipartimento di Matematica, Universit` a della Calabria - Via Ponte Bucci, cubo 30 B Arcavacata di Rende (Cs), Italy ( 2 ) INFN-Gruppo c. Cosenza - Cosenza, Italy ( 3 ) Dipartimento di Matematica e Informatica, Universit` a di Catania - Viale A. Doria 6 I-95125 Catania, Italy (ricevuto il 30 Ottobre 2009; approvato il 14 Gennaio 2010; pubblicato online il 16 Marzo 2010) Summary. — A hydrodynamic subband model for semiconductors is formulated by closing the moment system derived from the Schr¨odinger-Poisson-Boltzmann equations on the basis the maximum entropy principle. Explicit closure relations for fluxes and production terms are obtained taking into account scattering of electrons with acoustic and nonpolar optical phonons and surface scattering. PACS 73.63.-b – Electronic transport in nanoscale materials and structures. PACS 05.60.Gg – Quantum transport. PACS 73.63.Hs – Quantum wells. 1. – Quantum confinement By shrinking the dimension of electronic devices, effects of quantum confinement are observed, e.g., in MOSFETs the gate voltage and the energy barrier at the Si-SiO 2 interface confine the carriers near the oxide-silicon interface. Similarly in double-gate MOSFETs the potential between the two gates confines the electrons. The same effect is present in quantum wells in hetero-structures like AlGa-Ga. For a comprehensive review the reader is referred to [1, 2]. If electrons are quantized in a direction which we call z and free to move in the x-y plane, it is natural to assume the following ansatz for the wave function: ψ(k, r)= ψ(k x ,k y ,k z ,x,y,z)= 1 √ A ϕ(z)e ik || ·r || , with k || =(k x ,k y ) and r || =(x, y) denoting the longitudinal components of the wave- vector k and the position vector r, respectively, and A symbolizing the area of the xy cross-section. c  Societ` a Italiana di Fisica 155