Nonparabolic macroscopic transport models for device simulation based on bulk Monte Carlo data

Abstract

We derive higher-order macroscopic transport models for semiconductor device simulation from Boltzmann's transport equation using the method of moments. To obtain a tractable equation set suitable for numerical implementation the validity of the diffusion limit will be assumed which removes the convective terms from the equation system. The infinite hierarchy of equations is then truncated at the orders two ͑drift-diffusion model͒, four ͑energy-transport model͒, and six. Nonparabolicity correction factors are included in the streaming terms. Closure relations for the highest-order moments are obtained from a cold Maxwell distribution ͑drift-diffusion͒ and a heated Maxwell distribution ͑energy-transport͒. For the six moments model this issue is more complicated. In particular, this closure relation is identified to be crucial both in terms of accuracy and in terms of numerical stability. Various possible closure relations are discussed and compared. In addition to the closure of the highest-order moment, various transport parameters such as mobilities and relaxation times appear in the models and need to be accurately modeled. Particularly for higher-order transport models this is a complicated issue and since the analytical models used in our previous attempts did not deliver satisfactory results we extract all these parameters using homogeneous Monte Carlo simulations. Since all macroscopic transport models are based on rather stringent assumptions a practical evaluation is mandatory. Therefore, the proposed six moments model, a corresponding energy-transport model, and the drift-diffusion model are carefully compared to self-consistent Monte Carlo simulations.

Current transport models for nanoscale semiconductor devices

“…For simplicity we assume that the energy band is isotropic and parabolic, although generalization to a non-parabolic band is possible [35]. In order to derive the six-moments model, the following weight functions are chosen:…”

“…The energy-transport model requires for each carrier type the modeling of two mobilities for the current density and the energy flux, one relaxation time, and the non-parabilicity factor for non-parabolic bands [35]. Since these models are not always available, a simplified model with the energy flux mobility being proportional or equal to the carrier mobility has been proposed [23,43].…”

“…The scattering rates L mn ðk; k 0 Þ in (35) are related to the rates S mn ðk; k 0 Þ of the original Boltzmann equation via…”

A Hierarchy of Diffusive Higher-Order Moment Equations for Semiconductors

“…Together with the above condition, the kurtosis has to satisfy the inequality 3/5 ≤ β n ≤ 1 [22]. Clearly, β n = 1 corresponds to the energy-transport case for which λ 2 = 0.…”

Parameter modeling for higher-order transport models in UTB SOI MOSFETs

“…On an engineering level a very efficient way to find approximate solutions of BTE is the method of moments, in comparison to the very time consuming MC technique [2,3]. Considering the first two, four, and six moments, one can obtain the drift-diffusion, the energy-transport, and the six moments model respectively [4]. For these models, an M. Vasicek ( ) · J. Cervenka · M. Wagner · M. Karner · T. Grasser Institute for Microelectronics, TU Wien, Gußhausstraße 27-29, 1040 Vienna, Austria e-mail: vasicek@iue.tuwien.ac.at accurate description of transport parameters like the carrier mobility in the drift-diffusion model is very important.…”