Exact method to solve the Boltzmann equation to any order in the driving forces: Application to transport parameters

Abstract

In this paper a method to compute transport parameters in semiconductor devices as a function of the driving forces is presented. The method is based on an exact expansion of the distribution function (solution of the Boltzmann transport equation) in a series in the electric field and the gradient of the quasi‐Fermi potential. Eventhough this series probably diverges for high fields, it is still usable for low and moderate field intensities. Results for silicon clearly show that the usual parametrization of the mobility in TCAD models is inaccurate in pn junctions. Furthermore, it gives new insights into the form of the correct parametrization. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Coupling the Monte-Carlo method with semi-analytical solutions of the Boltzmann transport equation

“…(2) shows that the preponderate DF are the gradient of the quasi-Fermi potential, the electric field, and surprisingly the second derivative of the quasi-Fermi potential. In Section III it is shown that in small N IN structures the second derivative of the quasi-Fermi potential dominates, with the unfortunate consequence, that in such structures no natural TCAD mobility (in the sense of [1]) can be defined anymore.…”

“…2is valid, a conservative criterion has been used: TP are not allow to change by more than 25% of their equilibrium value. Using the notation of [1], this criterion can be mathematically written as…”

“…the electric field times the gradient of the quasi-Fermi potential as can be seen in Fig. 4 (M 1 , M 2 , M 3 , and M 4 are defined in [1]). Fig.…”

“…However, at low and medium electric fields and in moderately to highly doped regions the MC method is rather innefficient due to the very short mean free paths of the simulated particles. In [1] a semianalytical method has been described which allows to solve the BTE in all orders in the driving forces (DF). This semianalytical method can be coupled with the MC technique and can advantageously replace the MC method in the problematic regions.…”

Coupled Solution of Boltzmann Transport Equation, Maxwell’s and Navier Stokes equations

“…In a state of equilibrium a gas of particles has uniform composition with constant temperature and density. If the gas is subjected to a temperature difference or disturbed by externally applied electric, magnetic, or mechanical forces, it will be set in motion and the temperature, density, and composition may become functions of position and time, in other words, the gas moves out of equilibrium [6][7][8][9]. The Boltzmann Equation applies to a quantity known as the distribution function, which describes this non-equilibrium state mathematically and specifies how quickly and in what manner the state of the gas changes when the disturbing forces are varied.…”