Kinetic theory of thermotransport of polar semiconductors: Degenerate limit

Abstract

We develop a kinetic theory approach from the semiclassical Boltzmann transport equation for the thermotransport of electrons in degenerate polar semiconductors. The method of moments applied to the Boltzmann equation gives us a set of hydrodynamical equations which are closed up to thirteen relevant variables, including energy density, the stress tensor and the heat flux in the description. The closure of the balance equations is achieved by evaluating the higher order momenta, as well as the production terms, through a non equilibrium distribution function coming from the maximum entropy principle. We assume that electronoptical polar phonon interaction is the leading scattering process in order to obtain analytical expressions for both, the characteristic relaxation times and the usual thermoelectric coefficients. We also show that in this case the Onsager symmetry relationship is not satisfied Introduction Kinetic theoretical approaches have been frequently used to describe the transport properties of a semiclassical electron gas in the actually existing high integration semiconductor devices. To extend the usual drift diffusion model into the mesoscopic region, we need to include in the description some additional number of relevant variables. Those are chosen to describe the macroscopic state of electrons in the carrier transport process. The usual procedure to obtain a semiconductor generalized hydrodynamics scheme consists in constructing a hierarchy of balance equations, for the complete set of relevant variables, starting from the semiclassical Boltzmann transport equation(BTE) [1]. These hydrodynamic type equations constitute a closed set of balance equations for the relevant variables, once the higher order moments and the production terms have been completely evaluated.A well established method to calculate the higher order moments, in terms of the relevant local variables, takes the description of the semiconductor electron transport in terms of the evolution of a nonequilibrium distribution function coming from the general principle of entropy maximization [2]. Generally, the non equilibrium distribution has the form of a local equilibrium distribution function modified by a factor that depends on the relevant variables. For example, in degenerate semiconductors a local equilibrium Fermi-Dirac distribution function must be used as a weight function in the expansion. Another additional complication is present when we evaluate the production terms corresponding to the nonconserved variables [3]. This problem arises because of the intrinsic complexity of the collisional kernel in the BTE and is enhanced through the microscopic semiconductor scattering processes of conduction electrons which involve a variety of crystal lattice perturbations. That is the reason why in many cases, the closure problem is solved by means of the study of some phenomenological models, such as the relaxation time approximation or the computational Monte Carlo analysis. Under such circumstances the result is an hybrid scheme ...